# Anger function

In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as

${\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta }$

and is closely related to Bessel functions.

The Weber function (also known as Lommel-Weber function), introduced by H. F. Weber (1879), is a closely related function defined by

${\displaystyle \mathbf {E} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(\nu \theta -z\sin \theta )\,d\theta }$

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

{\displaystyle {\begin{aligned}\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{aligned}}}

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.

## Differential equations

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

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

More precisely, the Anger functions satisfy the equation

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

and the Weber functions satisfy the equation

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