ABSTRACT
Solid harmonics represent a complete solution of Laplace's equation in spherical polar coordinates, and serve as basis functions for the multipole expansion of a 3D potential function. Depending on their value at the origin of the coordinate system, the solid harmonics are either regular or irregular. Recurrence relations derived in this chapter provide numerically stable recipes for both types. When multiplied by charge strength, solid harmonics produce regular and irregular multipole moments, which transform according to the addition theorems of solid harmonics. Generating functions for solid harmonics lead naturally to an algebraic derivation of the addition theorems. This chapter explains the derivation of M2M, M2L, and L2L multipole operations and provides a simple vector diagram to visualize the origin translation of a multipole expansion.
