View source for Anger function

{{MSC|34}}
{{TEX|done}}

The function
\begin{equation}\label{e:Anger}
{\bf J}_\nu (x) = \frac{1}{\pi} \int_0^\pi \cos\, (\nu \theta - x \sin \theta)\, d\theta
\end{equation}
where  $x$ is a complex variable and $\nu$ a complex parameter. The functions are named after C. T. Anger who in {{Cite|An}} studied the integral on the right hand side of \eqref{e:Anger} when the upper limit is $2\pi$ rather than $\pi$ . The Anger function satisfies the inhomogeneous  [[Bessel equation|Bessel equation]]
\[
x^2 y'' + xy' + (x^2 - \nu^2) y = \frac{(z-\nu) \sin \nu\pi}{\pi} \, 
\]
(see 10.12 in {{Cite|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 {{Cite|Wa}}).

====References====
{|
|-
|valign="top"|{{Ref|An}}||valign="top"| C.T. Anger,   ''Neueste Schr. d. Naturf. d. Ges. i. Danzig'' , '''5'''  (1855)  pp. 1–29
|-
|valign="top"|{{Ref|Wa}}||valign="top"|  G.N.  Watson,   "A   treatise on the theory of Bessel functions", '''1–2''',  Cambridge Univ.   Press  (1952)  {{MR|1349110}} {{MR|1570252}}  {{MR|0010746}}   {{MR|1520278}}     {{ZBL|0849.33001}}  {{ZBL|0174.36202}}   {{ZBL|0063.08184}} 
|-
|}