Anger function

In mathematics, the Anger function, introduced by, 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 $$\textit{z}$$. It is closely related to the Bessel functions.

The Weber function (also known as Lommel–Weber function), introduced by, 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|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]]$$

\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 Jν are the same as Bessel functions Jν, and Weber functions can be expressed as finite linear combinations of Struve functions.

Power series expansion
The Anger function has the power series expansion


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

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


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