Anger function

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

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

with complex parameter ${\displaystyle \nu }$ and complex variable ${\displaystyle {\textit {z}}}$.^[1] It is closely related to the 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.