Weber function

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

The function \[ {\bf E}_\nu (x) = \frac{1}{\pi} \int_0^\pi \sin\, (\nu \theta - x \sin \theta)\, d\theta \] where $x$ is a complex variable and $\nu$ a complex parameter, first studied by Weber in [We]. The Weber function satisfies the inhomogeneous Bessel equation \[ x^2 y'' + xy' + (x^2 - \nu^2) y = - \frac{1}{\pi} \left( x+ \nu + (x-\nu) \cos \nu\pi\right)\, \] (see 10.12 in [Wa]).

When the parameter $\nu$ is not an integer, the Weber function has the following expansion: \[ {\bf E}_\nu (x) = \frac{1-\cos \nu\pi}{\nu \pi} \left[ 1 - \frac{x^2}{2^2-\nu^2} + \frac{x^4}{(2^2-\nu^2)(4^2 - \nu^2)} + \ldots \right] - \frac{1-\cos \nu\pi}{\pi}\left[ \frac{x}{1-\nu^2} - \frac{x^3}{(1-\nu^2) (3^2-\nu^2)} + \ldots \right] \] If $|x|$ is large and $|{\rm arg}\, x|< \frac{\pi}{2}$, the following asymptotic expansion is valid: \[ {\bf E_\nu} (x) = - Y_\nu (x) - \frac{1+\cos \nu\pi}{\pi x} \left[1 - \frac{1-\nu^2}{x^2} + \frac{(1-\nu^2)(3^2-\nu^2)}{x^4} + \ldots \right] - \nu \frac{1-\cos \nu\pi}{\pi x} \left[ \frac{1}{x} - \frac{2^2 - \nu^2}{x^3} + \frac{(2^2-\nu^2)(4^2-\nu^2)}{x^5} + \ldots \right]\, . \] where $Y_\nu$ is the Neumann function. If $\nu$ is not an integer, the Weber function is related to the Anger function ${\bf J}_\nu$ 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

[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
[We]	H.F. Weber, Zurich Vierteljahresschrift , 24 (1879) pp. 33–76