Bessel function of the first kind ($J_{\nu }(z)$)

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 $\mathrm{Re}\nu >-1/2$, it has the Poisson integral representation

For fixed $\nu$, putThenas $|z|\to \infty$, uniformly for $|\arg z|\le \pi -\delta$. In particular,