ABSTRACT
The associated Legendre functions are important mathematical functions that represent a special-case solution of Laplace's equations in spherical-polar coordinates. The most straightforward technique for obtaining associated Legendre functions of the m-th order is by differentiating Legendre polynomials of degree l, m times, where 0 ≤ m ≤ l. Associated Legendre functions are orthonormal, have symmetry in the azimuthal index m, and serve as the polar components of basis functions in multipole expansion series. This chapter gives a solid foundation for understanding these functions, which includes proofs of their orthonormality and symmetry properties, a derivation of numerically stable recurrence relations for efficiently computing the associated Legendre functions and an analytic derivation of their derivatives.
