Lamé function

Summary

A Lamé function is a solution of Lamé's equation $w'' + \left( h - \nu(\nu + 1) k^2 \operatorname{sn}^2(z, k) \right) w = 0$, where $\operatorname{sn}$ is a Jacobi elliptic function. The equation arises when Laplace's equation is separated in ellipsoidal coordinates, and its solutions are also called ellipsoidal harmonics. When $\nu = n$ is a nonnegative integer, special values of $h$ give Lamé polynomials, which play for ellipsoids the role that Legendre polynomials play for spheres. The equation is a special case of Heun's equation and a periodic Schrödinger equation with an elliptic potential.