Anger function

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

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

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

with complex parameter $\nu$ and complex variable $\textit{z}$ .{{springer|id=A/a012490|title=Anger function|first=A.P.|last= Prudnikov|authorlink=Anatolii Platonovich Prudnikov}} It is closely related to the Bessel functions.

The Weber function (also known as 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

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

and is closely related to Bessel functions of the second kind.

Relation between Weber and Anger functions

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 $\begin{align}
\sin(\pi \nu)\mathbf{J}_\nu(z) &= \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z), \\
-\sin(\pi \nu)\mathbf{E}_\nu(z) &= \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z),
\end{align}$

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 functions.

Power series expansion

The Anger function has the power series expansion{{dlmf|id=11.10|title=Anger-Weber Functions|first=R. B. |last=Paris}}

: $\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}.$

While the Weber function has the power series expansion

: $\mathbf{E}_\nu(z)=\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}-\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}.$

Differential equations

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

: $z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0 .$

More precisely, the Anger functions satisfy the equation

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

and the Weber functions satisfy the equation

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

Recurrence relations

The Anger function satisfies this inhomogeneous form of recurrence relation

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

While the Weber function satisfies this inhomogeneous form of recurrence relation

: $z\mathbf{E}_{\nu-1}(z)+z\mathbf{E}_{\nu+1}(z)=2\nu\mathbf{E}_\nu(z)-\frac{2(1-\cos\pi\nu)}{\pi}.$

Delay differential equations

The Anger and Weber functions satisfy these homogeneous forms of delay differential equations

: $\mathbf{J}_{\nu-1}(z)-\mathbf{J}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z),$

: $\mathbf{E}_{\nu-1}(z)-\mathbf{E}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z).$

The Anger and Weber functions also satisfy these inhomogeneous forms of delay differential equations

: $z\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z)\pm\nu\mathbf{J}_\nu(z)=\pm z\mathbf{J}_{\nu\mp1}(z)\pm\frac{\sin\pi\nu}{\pi},$

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

References

{{AS ref|12|498}}
C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
{{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. 33–76

Category:Special functions