 Legendre functions are solutions of Legendre's differential equation (generalized or not) with "non-integer" parameters.In physical settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation of variables in spherical coordinates.From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics, of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis.