Anger function

2020 Mathematics Subject Classification: Primary: 34-XX [MSN][ZBL]

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 [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 \[ x^2 y'' + xy' + (x^2 - \nu^2) y = \frac{(z-\nu) \sin \nu\pi}{\pi} \, \] (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]).

References

[An]	C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig , 5 (1855) pp. 1–29
[Wa]	G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) MR1349110 MR1570252 MR0010746 MR1520278 Zbl 0849.33001 Zbl 0174.36202 Zbl 0063.08184