ABSTRACT
Spherical harmonics are solutions of Laplace's equation that combine both polar and azimuthal components, and can be used in describing the electrostatic potential on the surface of a unit sphere. Although both the spherical harmonics and the special-case associated Legendre functions form a complete orthonormal set, only the spherical harmonics can be used as basis functions for a series expansion of arbitrary functions, including electrostatic potential in the Fast Multipole Method. Deriving the orthogonality and normalization conditions followed by an analysis of their symmetry properties and recurrence relations provides a detailed mathematical description of the spherical harmonics. These fall into the groups of zonal, sectorial, and tesseral or real functions, depending on the values of the orbital and azimuthal indices. A visual representation of the nodal properties of some of the spherical harmonics provides an insight into their spatial characteristics.
