Legendre functions

Summary

Legendre functions are the solutions of the general Legendre differential equation, which has two parameters, a degree $\nu$ and an order $\mu$. The solutions of the first kind, $P_\nu^\mu$, and of the second kind, $Q_\nu^\mu$, are the standard basis of solutions. When $\mu = 0$ and $\nu$ is a nonnegative integer, the first kind gives the Legendre polynomials. They arise when the Laplace equation or the Helmholtz equation is separated in spherical coordinates.