Hansen-Bessel Formula

$$J_n(z) = {1\over 2\pi}\int_{-\pi}^\pi e^{iz\cos t}e^{in(t-\pi/2)}\,dt$$

$$= {i^{-n}\over\pi} \int_0^\pi e^{iz\cos t}cos(nt)\,dt$$

$$= {1\over\pi}\int_0^\pi \cos(z\sin t-nt)\,dt$$

for $n=0$, 1, 2, ..., where $J_n(z)$ 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.