SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Anger function</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Anger function</span></span>

</h1>

</header>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>Anger function</b>, introduced by <a href="Carl_Theodor_Anger" title="Carl Theodor Anger">C. T. Anger</a> (<a href="#CITEREFAnger1855">1855</a>), is a function defined as

</p>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta }</annotation>

</semantics>

</math></span><img src="./da70c525edee6a52fd7a6a7428dd927b3afaf7b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.646ex; height:5.843ex;" alt="{\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta }" loading="lazy"></span></dd></dl>

<p>with complex parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>

</semantics>

</math></span><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"></span> and complex variable <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textit {z}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\textit {z}}}</annotation>

</semantics>

</math></span><img src="./1a20ef8e203e8dc84188395d8b6deb8328942552.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.132ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle {\textit {z}}}" loading="lazy"></span>.<sup id="cite_ref-EOM_Anger_1-0" class="reference"><a href="#cite_note-EOM_Anger-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is closely related to the <a href="Bessel_function" title="Bessel function">Bessel functions</a>.

</p><p>The <b>Weber function</b> (also known as <b><a href="Eugen_von_Lommel" title="Eugen von Lommel">Lommel</a>–Weber function</b>), introduced by <a href="Heinrich_Friedrich_Weber" title="Heinrich Friedrich Weber">H. F. Weber</a> (<a href="#CITEREFWeber1879">1879</a>), is a closely related function defined by

</p>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(\nu \theta -z\sin \theta )\,d\theta }</annotation>

</semantics>

</math></span><img src="./5b0ad921e335d7cfe2426f201934272b2b8d4cfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.767ex; height:5.843ex;" alt="{\displaystyle \mathbf {E} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(\nu \theta -z\sin \theta )\,d\theta }" loading="lazy"></span></dd></dl>

<p>and is closely related to <a href="Bessel_function" title="Bessel function">Bessel functions</a> of the second kind.

</p>

<div class="mw-heading mw-heading2"><h2 id="Relation_between_Weber_and_Anger_functions">Relation between Weber and Anger functions</h2></div>

<p>The Anger and Weber functions are related by

</p>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin(\pi \nu )\mathbf {J} _{\nu }(z)&=\cos(\pi \nu )\mathbf {E} _{\nu }(z)-\mathbf {E} _{-\nu }(z),\\-\sin(\pi \nu )\mathbf {E} _{\nu }(z)&=\cos(\pi \nu )\mathbf {J} _{\nu }(z)-\mathbf {J} _{-\nu }(z),\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./ec9d71d0eb77a7a253b8e96705a3fca220190761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.798ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\sin(\pi \nu )\mathbf {J} _{\nu }(z)&=\cos(\pi \nu )\mathbf {E} _{\nu }(z)-\mathbf {E} _{-\nu }(z),\\-\sin(\pi \nu )\mathbf {E} _{\nu }(z)&=\cos(\pi \nu )\mathbf {J} _{\nu }(z)-\mathbf {J} _{-\nu }(z),\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>so in particular if ν is not an <a href="Integer" title="Integer">integer</a> they can be expressed as linear combinations of each other. If ν is an integer then Anger functions <b>J</b><sub>ν</sub> are the same as Bessel functions <i>J</i><sub>ν</sub>, and Weber functions can be expressed as finite linear combinations of <a href="Struve_function" title="Struve function">Struve functions</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Power_series_expansion">Power series expansion</h2></div>

<p>The Anger function has the <a href="Power_series" title="Power series">power series</a> expansion<sup id="cite_ref-DLMF_2-0" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{\nu }(z)=\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}+\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}.}</annotation>

</semantics>

</math></span><img src="./a1c390444ca1ee123c57fba223004d1db8c78a38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:100.847ex; height:8.509ex;" alt="{\displaystyle \mathbf {J} _{\nu }(z)=\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}+\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}.}" loading="lazy"></span></dd></dl>

<p>While the Weber function has the power series expansion<sup id="cite_ref-DLMF_2-1" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{\nu }(z)=\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}-\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}.}</annotation>

</semantics>

</math></span><img src="./0a848657b0e1136451cb7749aa00a70d7352ad09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:101.224ex; height:8.509ex;" alt="{\displaystyle \mathbf {E} _{\nu }(z)=\sin {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k}}{4^{k}\Gamma \left(k+{\frac {\nu }{2}}+1\right)\Gamma \left(k-{\frac {\nu }{2}}+1\right)}}-\cos {\frac {\pi \nu }{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k+1}}{2^{2k+1}\Gamma \left(k+{\frac {\nu }{2}}+{\frac {3}{2}}\right)\Gamma \left(k-{\frac {\nu }{2}}+{\frac {3}{2}}\right)}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Differential_equations">Differential equations</h2></div>

<p>The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=0.}</annotation>

</semantics>

</math></span><img src="./73e1ae5736b69142ffb4165d2fc01a5a0152fe9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.225ex; height:3.176ex;" alt="{\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=0.}" loading="lazy"></span></dd></dl>

<p>More precisely, the Anger functions satisfy the equation<sup id="cite_ref-DLMF_2-2" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y={\frac {(z-\nu )\sin(\pi \nu )}{\pi }},}</annotation>

</semantics>

</math></span><img src="./2ab5084f786887789ced9dc487121626d4a1a27d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.485ex; height:5.676ex;" alt="{\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y={\frac {(z-\nu )\sin(\pi \nu )}{\pi }},}" loading="lazy"></span></dd></dl>

<p>and the Weber functions satisfy the equation<sup id="cite_ref-DLMF_2-3" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=-{\frac {z+\nu +(z-\nu )\cos(\pi \nu )}{\pi }}.}</annotation>

</semantics>

</math></span><img src="./53de6efec20a89310b17cdeb2dc2ab521d9a6d67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:52.55ex; height:5.676ex;" alt="{\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=-{\frac {z+\nu +(z-\nu )\cos(\pi \nu )}{\pi }}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Recurrence_relations">Recurrence relations</h2></div>

<p>The Anger function satisfies this inhomogeneous form of <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a><sup id="cite_ref-DLMF_2-4" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\mathbf {J} _{\nu -1}(z)+z\mathbf {J} _{\nu +1}(z)=2\nu \mathbf {J} _{\nu }(z)-{\frac {2\sin \pi \nu }{\pi }}.}</annotation>

</semantics>

</math></span><img src="./30d8664b4b4973b6dd585a98fd6ef8b0c4c5c978.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.536ex; height:5.176ex;" alt="{\displaystyle z\mathbf {J} _{\nu -1}(z)+z\mathbf {J} _{\nu +1}(z)=2\nu \mathbf {J} _{\nu }(z)-{\frac {2\sin \pi \nu }{\pi }}.}" loading="lazy"></span></dd></dl>

<p>While the Weber function satisfies this inhomogeneous form of <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a><sup id="cite_ref-DLMF_2-5" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\mathbf {E} _{\nu -1}(z)+z\mathbf {E} _{\nu +1}(z)=2\nu \mathbf {E} _{\nu }(z)-{\frac {2(1-\cos \pi \nu )}{\pi }}.}</annotation>

</semantics>

</math></span><img src="./f38d52f3636ade624313fcfe6e1062228d86bfa8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.345ex; height:5.676ex;" alt="{\displaystyle z\mathbf {E} _{\nu -1}(z)+z\mathbf {E} _{\nu +1}(z)=2\nu \mathbf {E} _{\nu }(z)-{\frac {2(1-\cos \pi \nu )}{\pi }}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Delay_differential_equations">Delay differential equations</h2></div>

<p>The Anger and Weber functions satisfy these homogeneous forms of <a href="Delay_differential_equation" title="Delay differential equation">delay differential equations</a><sup id="cite_ref-DLMF_2-6" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{\nu -1}(z)-\mathbf {J} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z),}</annotation>

</semantics>

</math></span><img src="./584d3522a68edf64a5b970a8c82f91c751dd8f86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.337ex; height:5.509ex;" alt="{\displaystyle \mathbf {J} _{\nu -1}(z)-\mathbf {J} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z),}" loading="lazy"></span></dd>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{\nu -1}(z)-\mathbf {E} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {E} _{\nu }(z).}</annotation>

</semantics>

</math></span><img src="./96e19ef960587d2d307ef35dfb1795e9caab308b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.465ex; height:5.509ex;" alt="{\displaystyle \mathbf {E} _{\nu -1}(z)-\mathbf {E} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {E} _{\nu }(z).}" loading="lazy"></span></dd></dl>

<p>The Anger and Weber functions also satisfy these inhomogeneous forms of <a href="Delay_differential_equation" title="Delay differential equation">delay differential equations</a><sup id="cite_ref-DLMF_2-7" class="reference"><a href="#cite_note-DLMF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z)\pm \nu \mathbf {J} _{\nu }(z)=\pm z\mathbf {J} _{\nu \mp 1}(z)\pm {\frac {\sin \pi \nu }{\pi }},}</annotation>

</semantics>

</math></span><img src="./926c1e83333aaeecad65948aec46a4622013b8ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:42.774ex; height:5.509ex;" alt="{\displaystyle z{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z)\pm \nu \mathbf {J} _{\nu }(z)=\pm z\mathbf {J} _{\nu \mp 1}(z)\pm {\frac {\sin \pi \nu }{\pi }},}" loading="lazy"></span></dd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z{\dfrac {\partial }{\partial z}}\mathbf {E} _{\nu }(z)\pm \nu \mathbf {E} _{\nu }(z)=\pm z\mathbf {E} _{\nu \mp 1}(z)\pm {\frac {1-\cos \pi \nu }{\pi }}.}</annotation>

</semantics>

</math></span><img src="./1a7800c1bfda69656008f4aea5d03c933d1c7d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:48.161ex; height:5.509ex;" alt="{\displaystyle z{\dfrac {\partial }{\partial z}}\mathbf {E} _{\nu }(z)\pm \nu \mathbf {E} _{\nu }(z)=\pm z\mathbf {E} _{\nu \mp 1}(z)\pm {\frac {1-\cos \pi \nu }{\pi }}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFPrudnikov2001" class="citation cs2"><a href="Anatolii_Platonovich_Prudnikov" class="mw-redirect" title="Anatolii Platonovich Prudnikov">Prudnikov, A.P.</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Anger_function">"Anger function"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></span>

</li>

<li id="cite_note-DLMF-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-DLMF_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-DLMF_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-DLMF_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-DLMF_2-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-DLMF_2-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-DLMF_2-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-DLMF_2-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-DLMF_2-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFParis2010" class="citation cs2">Paris, R. B. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/11.10">"Anger-Weber Functions"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <i><a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a></i>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</span>

</li>

</ol></div></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}

/* end https://en.wikipedia.org/ */

</style><div class="refbegin" style="">

<ul><li><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_498.htm">"Chapter 12"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 498. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite></li>

<li>C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29</li>

<li><cite id="CITEREFPrudnikov2001" class="citation cs2">Prudnikov, A.P. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Weber_function">"Weber function"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><a href="G.N._Watson" class="mw-redirect" title="G.N. Watson">G.N. Watson</a>, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952)</li>

<li>H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76</li></ul>

</div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-12" href="https://en.wikipedia.org/wiki/?title=Anger_function&oldid=1285161811">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>