Poisson's Bessel Function Formula

For $\Re[\nu]>-1/2$,

$$J_\nu(z)=\left({z\over 2}\right)^\nu{2\over\sqrt{\pi}\,\Gamm... ...style{1\over 2}})}\int_0^{\pi/2}\cos(z\cos t)\sin^{2\nu}t\,dt,$$

where $J_\nu(z)$ is a Bessel Function of the First Kind, and $\Gamma(z)$ is the Gamma Function.

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1472, 1980.