Anger function: Difference between revisions

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[[File:Plot of the Anger function J v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Anger function J v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Anger function J v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]

In [[mathematics]], the '''Anger function''', introduced by {{harvs|txt|authorlink=Carl Theodor Anger|first=C. T.|last=Anger|year=1855}}, is a function defined as

: <math>\mathbf{J}_\nu(z)=\frac{1}{\pi} \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta</math>

with complex parameter <math>\nu</math> and complex variable <math>\textit{z}</math>.<ref name="EOM_Anger">{{springer|id=A/a012490|title=Anger function|first=A.P.|last= Prudnikov|authorlink=Anatolii Platonovich Prudnikov}}</ref> It is closely related to the [[Bessel function]]s.

and is closely related to [[Bessel function]]s.

The '''Weber function''' (also known as '''[[Eugen von Lommel|Lommel]]–Weber function'''), introduced by {{harvs|txt|authorlink=Heinrich Friedrich Weber|first=H. F.|last=Weber|year=1879}}, is a closely related function defined by

: <math>\mathbf{E}_\nu(z)=\frac{1}{\pi} \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta</math>

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The Anger and Weber functions are related by

:[[File:Plot of the Weber function E v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the Weber function E v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the Weber function E v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]<math>

:<math>\sin(\pi \nu)\mathbf{J}_\nu(z) = \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z)</math>

\begin{align}

:<math>\sin(\pi \nu)\mathbf{J}_\nu(z) &= \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z)</math>, \\

:<math>-\sin(\pi \nu)\mathbf{E}_\nu(z) &= \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z)</math>,

\end{align}

</math>

so in particular if &nu;ν is not an [[integer]] they can be expressed as linear combinations of each other. If &nu;ν is an integer then Anger functions '''J'''_&nu;ν are the same as Bessel functions ''J''_&nu;ν, and Weber functions can be expressed as finite linear combinations of [[Struve function]]s.

:<math>-\sin(\pi \nu)\mathbf{E}_\nu(z) = \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z)</math>

==Power series expansion==

so in particular if &nu; is not an integer they can be expressed as linear combinations of each other. If &nu; is an integer then Anger functions '''J'''_&nu; are the same as Bessel functions ''J''_&nu;, and Weber functions can be expressed as finite linear combinations of [[Struve function]]s.

The Anger function has the [[power series]] expansion<ref name=DLMF>{{dlmf|id=11.10|title=Anger-Weber Functions|first=R. B. |last=Paris}}</ref>

:<math>\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{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)^kz^{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)}.</math>

While the Weber function has the power series expansion<ref name=DLMF/>

:<math>\mathbf{E}_\nu(z)=\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{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)^kz^{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)}.</math>

==Differential equations==

The Anger and Weber functions are solutions of inhomogenousinhomogeneous forms of Bessel's equation

:<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0 .</math>. More precisely,

the Anger functions satisfy the equation

More precisely, the Anger functions satisfy the equation<ref name=DLMF/>

:<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = \frac{(z-\nu)\sin(\pi \nu)}{\pi} ,</math>

and the AngerWeber functions satisfy the equation<ref name=DLMF/>

:<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = -\frac{z+\nu+(z-\nu)\cos(\pi \nu)}{\pi}.</math>

==Recurrence relations==

The Anger function satisfies this inhomogeneous form of [[recurrence relation]]<ref name=DLMF/>

:<math>z\mathbf{J}_{\nu-1}(z)+z\mathbf{J}_{\nu+1}(z)=2\nu\mathbf{J}_\nu(z)-\frac{2\sin\pi\nu}{\pi}.</math>

While the Weber function satisfies this inhomogeneous form of [[recurrence relation]]<ref name=DLMF/>

:<math>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}.</math>

==Delay differential equations==

The Anger and Weber functions satisfy these homogeneous forms of [[delay differential equation]]s<ref name=DLMF/>

:<math>z^2y^\mathbf{J}_{\primenu-1}(z)-\primemathbf{J} + zy^_{\prime nu+1}(z^)=2-\nu^2)ydfrac{\partial}{\partial = (z-}\nu)mathbf{J}_\sinnu(\pi z)/\pi,</math>

:<math>\mathbf{E}_{\nu-1}(z)-\mathbf{E}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z).</math>

andThe theAnger and Weber functions also satisfy thethese inhomogeneous forms of [[delay differential equation]]s<ref name=DLMF/>

:<math>z^2y^\dfrac{\prime\primepartial} + zy^{\primepartial +(z^2-}\mathbf{J}_\nu^2)y = -((z+)\pm\nu\mathbf{J}_\nu(z)=\pm + (z-\mathbf{J}_{\nu)\cosmp1}(\pi z))/\pm\frac{\sin\pi.\nu}{\pi},</math>

:<math>z\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z)\pm\nu\mathbf{E}_\nu(z)=\pm z\mathbf{E}_{\nu\mp1}(z)\pm\frac{1-\cos\pi\nu}{\pi}.</math>

==References==

{{Reflist}}

{{Refbegin}}

*{{AS ref|12|498}}

*C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig , 5 (1855) pp. &nbsp;1–29

*{{springer|id=A/a012490|title=Anger function|first=A.P.|last= Prudnikov}}

*{{springer|id=W/w097320|title=Weber function|first=A.P.|last= Prudnikov}}

*[[G.N. Watson]], "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952)

*H.F. Weber, Zurich Vierteljahresschrift , 24 (1879) pp. &nbsp;33–76

{{Refend}}

[[Category:Special functions]]