ABSTRACT
Solutions of Laplace's equation known as the Legendre polynomials are useful in describing electrostatic potential functions in regions of space that are far from the charge sources. In particular, these solutions can assist in the factorization of the inverse distance function, an essential prerequisite for replacing the conventional quadratic method of calculating electrostatic interactions with one that scales linearly. Of their nature, the Legendre polynomials form a complete orthonormal set and, as such, are ideally suited for converting the electrostatic potential function into a series expansion. Working efficiently with Legendre polynomials involves making extensive use of recurrence relations and this makes them a powerful tool at the center of the Fast Multipole Method. This chapter presents a detailed mathematical derivation of the key features of the Legendre polynomials.
