Two-dimensional harmonic and Green's functions on a spherical surface

Abstract

The solutions of the Laplace-Beltrami equation on a spherical surface are constructed by the method of separation of variables, as the products of the Fourier basis functions of the azimuthal angle and the integer powers of tangent or cotangent functions of half the polar angle. The Legendre operator acting on the latter functions yields zero. The construction of the Green's function as the solution of the corresponding Poisson-Beltrami equation with a unit point source on the spherical surface is also constructed using the two-dimensional spherical harmonic basis.