Schläfli's Formula

For $\Re[z]>0$,

$$J_\nu(z)={1\over\pi}\int_0^{\pi/2} \cos(z\sin t-\nu t)\,dt-{\sin(\nu\pi)\over\pi}\int_0^\infty e^{-z\sinh t}e^{-\nu t}\,dt,$$

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

References

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