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.

What are the spherical harmonics used for? They are most often used to create an orthonormal basis for a function on the unit sphere. This is because a function on the sphere with bandwidth L can be represented as a sum of the spherical harmonics of up to order L. A spherical fourier transform or wavelet transform is used to determine the spherical harmonic coefficients, which can be operated on (filtered) before the inverse transform. The most common filtering operation on the spherical harmonics is a triangular truncation, or high-pass filter, where harmonics at or above order N are eliminated.

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.