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-Weber|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>

\begin{align}

\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{align}

</math>

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

==Power series expansion==

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>

:<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>

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==

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

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 Weber 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>\mathbf{J}_{\nu-1}(z)-\mathbf{J}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z),</math>

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

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

:<math>z\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z)\pm\nu\mathbf{J}_\nu(z)=\pm z\mathbf{J}_{\nu\mp1}(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|authorlink=Anatolii Platonovich 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

{{ReflistRefend}}

[[Category:Special functions]]