Spheroidal wave function

Summary

Spheroidal wave functions are solutions of the equations obtained by separation of variables in the Helmholtz equation in prolate or oblate spheroidal coordinates. The angular spheroidal wave functions $S_{mn}(c, x)$ are the solutions bounded on $[-1, 1]$, and they exist only for special eigenvalues $\lambda_{mn}(c)$. For $c = 0$ they reduce to the associated Legendre polynomials, so they deform the spherical case to the spheroidal one. The prolate functions with $m = 0$ are also central in signal processing. There they are the functions of a given bandwidth that are most concentrated in a time interval.