spherical harmonics

A class of functions that are quite useful, especially in chemistry and physics. They are eigenfunctions of the angular momentum operator, among other things.

Spherical harmonics usually arise in the solution of Laplace's equation in spherical coordinates where azimuthal symmetry is not present. They satisfy the spherical harmonic equation, which is simply the angular part of Laplace's equation in spherical coordinates. These functions are given by:

Y^m_l(θ, φ) = sqrt(((2l+1)(l-m)!)/(4π(l+m)!))P^m_l(cos θ) exp(imφ)

where P^m_l(z) is an associated Legendre polynomial.

An example wave function having such properties are spherically expanding electromagnetic waves. Spherical harmonics are useful in performing near to far-field transforms on such wave functions, as well as filtering and interpolation.