View source for Weber function

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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 {{Cite|We}}. The term Weber function
is also used sometimes for the [[Cylinder functions|cylinder function]] of second type $Y_\nu$ (also called Bessel functions of second type or Neumann functions and denoted sometimes by $N_\nu$).

The Weber function satisfies the inhomogeneous [[Bessel equation|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 {{Cite|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|Neumann function]]. If $\nu$ is not an integer, the Weber function is related to the [[Anger function|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 {{Cite|Wa}}).

====References====
{|
|-
|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}} 
|-
|valign="top"|{{Ref|We}}||valign="top"| H.F. Weber,   ''Zurich Vierteljahresschrift'' , '''24'''  (1879)  pp. 33–76
|-
|}