Lommel–Weber Function

In mathematics, the Anger function, introduced by , is a function defined as

: $\mathbf_\nu(z)=\frac \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta$

with complex parameter $\nu$ and complex variable $\textit$. 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

: $\mathbf_\nu(z)=\frac \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta$

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

:$\begin \sin(\pi \nu)\mathbf_\nu(z) &= \cos(\pi\nu)\mathbf_\nu(z)-\mathbf_(z), \\ -\sin(\pi \nu)\mathbf_\nu(z) &= \cos(\pi\nu)\mathbf_\nu(z)-\mathbf_(z), \end$

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

:$\mathbf_\nu(z)=\cos\frac\sum_^\infty\frac+\sin\frac\sum_^\infty\frac.$

While the Weber function has the power series expansion

:$\mathbf_\nu(z)=\sin\frac\sum_^\infty\frac-\cos\frac\sum_^\infty\frac.$

Differential equations

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

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

More precisely, the Anger functions satisfy the equation

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

and the Weber functions satisfy the equation

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

Recurrence relations

The Anger function satisfies this inhomogeneous form of recurrence relation

:$z\mathbf_(z)+z\mathbf_(z)=2\nu\mathbf_\nu(z)-\frac.$

While the Weber function satisfies this inhomogeneous form of recurrence relation

:$z\mathbf_(z)+z\mathbf_(z)=2\nu\mathbf_\nu(z)-\frac.$

Delay differential equations

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

:$\mathbf_(z)-\mathbf_(z)=2\dfrac\mathbf_\nu(z),$
:$\mathbf_(z)-\mathbf_(z)=2\dfrac\mathbf_\nu(z).$

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

:$z\dfrac\mathbf_\nu(z)\pm\nu\mathbf_\nu(z)=\pm z\mathbf_(z)\pm\frac,$
:$z\dfrac\mathbf_\nu(z)\pm\nu\mathbf_\nu(z)=\pm z\mathbf_(z)\pm\frac.$

References

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