= Anger function
{c}
{title2=$\mathbf J_\nu(z)$}
The Anger function is the <entire function>
$$
\mathbf J_\nu(z)=\frac1\pi\int_0^\pi\cos(\nu\theta-z\sin\theta)\,d\theta.
$$
For integer order $n$ it equals the first-kind <Bessel function> $J_n(z)$. At noninteger order it generally differs from $J_\nu$, despite the similar notation.<NIST Digital Library of Mathematical Functions>[Section 11.10(i), equation 11.10.1, and the integer-order interrelations]{ref}
The distinction appears in its inhomogeneous differential equation:
$$
z^2y''+zy'+(z^2-\nu^2)y=\frac{(z-\nu)\sin\pi\nu}{\pi}.
$$
Indeed, differentiating under the integral and integrating the derivative of $(\nu+z\cos\theta)\sin(\nu\theta-z\sin\theta)$ reduces the left side to its endpoint values, giving the displayed forcing. At integer order the forcing vanishes, consistently with the ordinary <Bessel equation>. This integral representation also makes analyticity in both the argument and the order visible.
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