Fourier–Legendre series

Summary

A Fourier–Legendre series expands a function on $[-1, 1]$ as a sum of Legendre polynomials. It is the generalized Fourier series for the orthogonal system of Legendre polynomials in $L^2([-1, 1])$. Since this system is complete, the series of every square-integrable function converges to it in the mean-square sense. Such series appear when one solves the Laplace equation in spherical coordinates.