Bessel function of the first kind
= Bessel function of the first kind
{c}
{title2=$J_\nu(z)$}
The Bessel function of the first kind $J_\nu$ is the solution of the <Bessel differential equation> selected by its regular power-series behavior at the origin. For $\operatorname{Re}\nu>-1/2$, it has the Poisson integral representation
$$
J_\nu(z)=\frac{(z/2)^\nu}{\Gamma(\nu+1/2)\sqrt\pi}
\int_{-1}^{1}e^{izt}(1-t^2)^{\nu-1/2}\,dt.
$$