Spherical Harmonic Addition Theorem
Alternate name
Definition
A formula also known as the Legendre addition theorem which is derived by finding Green's functions for the spherical harmonic expansion and equating them to the generating function for Legendre polynomials. When γ is defined by cos γ congruent cos θ_1 cos θ_2 + sin θ_1 sin θ_2 cos(ϕ_1 - ϕ_2), The Legendre polynomial of argument γ is given by P_l(cos γ) | = | (4π)/(2l + 1) sum_(m = - l)^l (-1)^m Y_l^m(θ_1, ϕ_1) Y_l^(-m)(θ_2, ϕ_2) | = | (4π)/(2l + 1) sum_(m = - l)^l Y_l^m(θ_1, ϕ_1)Y^__l^m(θ_2, ϕ_2) | = | P_l(cos θ_1) P_l(cos θ_2) + 2 sum_(m = 1)^l ((l - m)!)/((l + m)!) P_l^m(cos θ_1) P_l^m(cos θ_2) cos[m(ϕ_1 - ϕ_2)].
Related terms
