Anger function

Relation between Weber and Anger functions

The Anger and Weber functions are related by : \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 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 : \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

• C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
• H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76

References

1. Prudnikov, A.P.. "Anger function".
2. Paris, R. B.. "Anger–Weber Functions".