The spherical harmonics form a complete orthogonal system, so an arbitrary real function f(θ, ϕ) can be expanded in terms of complex spherical harmonics by f(θ, ϕ) congruent sum_(l = 0)^∞ sum_(m = 0)^l A_l^m Y_l^m(θ, ϕ), or in terms of real spherical harmonics by f(θ, ϕ) congruent sum_(l = 0)^∞ sum_(m = 0)^l[C_l^m Y_l^m^c(θ, ϕ) + S_l^m Y_l^m^s(θ, ϕ)]. The representation of a function f(θ, ϕ) as such a double series is a generalized Fourier series known as a Laplace series.

complete orthogonal system | Fourier-Legendre series | Fourier series | generalized Fourier series | spherical harmonic

Pierre-Simon Laplace