Jacobi-Anger Expansion

$$e^{iz\cos\theta} = \sum_{n=-\infty}^\infty i^nJ_n(z)e^{in\theta},$$

where $J_n(z)$ is a Bessel Function of the First Kind. The identity can also be written

$$e^{iz\cos\theta}=J_0(z)+2\sum_{n=1}^\infty i^n J_n(z)\cos(n\theta).$$

This expansion represents an expansion of plane waves into a series of cylindrical waves.

See also Bessel Function of the First Kind