The Condon-Shortley phase is the factor of (-1)^m that occurs in some definitions of the spherical harmonics to compensate for the lack of inclusion of this factor in the definition of the associated Legendre polynomials. Using the Condon-Shortley convention in the definition of the spherical harmonic after omitting it in the definition of P_l^m(cos θ) gives Y_l^m(θ, ϕ) = (-1)^m sqrt((2l + 1)/(4π) ((l - m)!)/((l + m)!))P_l^m(cos θ) e^(i m ϕ) (Arfken 1985, p.

associated Legendre polynomial | spherical harmonic