The spherical harmonics Y_l^m(θ, ϕ) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notational convention being used. In this entry, θ is taken as the polar (colatitudinal) coordinate with θ element [0, π], and ϕ as the azimuthal (longitudinal) coordinate with ϕ element [0, 2π). This is the convention normally used in physics, as described by Arfken and the Wolfram Language (in mathematical literature, θ usually denotes the longitudinal coordinate and ϕ the colatitudinal coordinate). Spherical harmonics are implemented in the Wolfram Language as SphericalHarmonicY[l, m, theta, phi].

associated Legendre polynomial | Condon-Shortley phase | correlation coefficient | Laplace series | sectorial harmonic | solid harmonic | spherical harmonic addition theorem | spherical harmonic closure relations | spherical harmonic differential equation | surface harmonic | tesseral harmonic | vector spherical harmonic | zonal harmonic

SphericalHarmonicY