Potential Theory and Spherical Harmonics

Abstract

In this chapter we investigate solutions of the potential equation due to Laplace in the homogeneous case and due to Poisson in the inhomogeneous case. Parallel to the theory of holomorphic functions we develop the theory of harmonic functions annihilating the Laplace equation. By the ingenious Perron method we shall solve Dirichlet’s problem for harmonic functions. Then we present the theory of spherical harmonics initiated by Legendre and elaborated by Herglotz to the present form. This system of functions constitutes an explicit basis for the standard Hilbert space and simultaneously provides a model for the ground states of atoms.