Anger function

Remarks

Both are entire functions of $z$ and of $\nu$, since the integrands are entire and the interval is compact. If $n$ is an integer, then $\mathbf{J}_n = J_n$, the Bessel function of the first kind, by Bessel's integral. Example: at $z = 0$, $\mathbf{J}_\nu(0) = \frac{1}{\pi} \int_0^\pi \cos(\nu\tau)\, d\tau = \frac{\sin(\nu\pi)}{\nu\pi}$ for $\nu \ne 0$. A non-example: for $\nu = \frac{1}{2}$, $\mathbf{J}_{1/2}(0) = \frac{2}{\pi} \ne 0$, while $J_{1/2}(0) = 0$, so $\mathbf{J}_{1/2}$ is not the Bessel function $J_{1/2}$. The Anger function solves $z^2 w'' + z w' + (z^2 - \nu^2)\, w = \frac{(z - \nu)\sin(\nu\pi)}{\pi}$, so it solves Bessel's equation exactly when $\nu$ is an integer. The Weber function $\mathbf{E}_\nu$ here is different from the Bessel function of the second kind, which is also sometimes called a Weber function.