# Anger function

The function $$\tag{1} {\bf J}_\nu (x) = \frac{1}{\pi} \int_0^\pi \cos\, (\nu \theta - x \sin \theta)\, d\theta$$ where $x$ is a complex variable and $\nu$ a complex parameter. The functions are named after C. T. Anger who in [An] studied the integral on the right hand side of (1) when the upper limit is $2\pi$ rather than $\pi$ . The Anger function satisfies the inhomogeneous Bessel equation $x^2 y'' + xy' + (x^2 - \nu^2) y = \frac{(z-\nu) \sin \nu\pi}{\pi} \,$ (see 10.12 in [Wa]).
For integers $\nu =n$ the Anger function coincides with the Bessel function $J_\nu$ of order $n$ (cf. Bessel functions). For non-integer $\nu$ the following expansion is valid: ${\bf J}_\nu (x) = \frac{\sin \nu\pi}{\nu \pi} \left[ 1 - \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots \right] + \frac{\sin \nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right]$ For $|x|$ large and $|{\rm arg}\, x| < \pi$ we moreover have the asymptotic expansion ${\bf J_\nu} (x) = - J_\nu (x) + \frac{\sin \nu\pi}{\pi x} \left[1 - \frac{1-\nu^2}{x^2} + \frac{(1-\nu^2)(3^2-\nu^2)}{x^4} + \ldots \right] - \nu \frac{\sin \nu \pi}{\pi x} \left[ \frac{1}{x} - \frac{2^2 - \nu^2}{x^3} + \frac{(2^2-\nu^2)(4^2-\nu^2)}{x^5} + \ldots \right]\, .$ If $\nu$ is not an integer, the Anger function is related to the Weber function ${\bf E}_\nu$ by the following equations: \begin{align} & \sin \nu\pi\, {\bf J}_\nu (x) = \cos \nu \pi\, {\bf E}_\nu (x) - {\bf E}_{-\nu} (x)\\ & \sin \nu\pi\, {\bf E}_\nu (x) = {\bf J}_{-\nu} (x) - \cos \nu\pi\, {\bf J}_\nu (x)\, \end{align} (cf. 10.11 in [Wa]).