Integrals Pn(x)dx Pn+1(x)-Pn-1(x) (E170) x"P(x)dx=0, m<n (E171) x"P(x)d (2n+1)! (2k+2n+1)!(k-n)! Pn(x)dr= 2v2 r(n+1) (E.175) +2(2nn Fourier-Legendre series expansion of a function f(x)=∑anPn(x),-1≤x≤ (E177) f(r)Pn(x)d E3 Spherical harmonics Notation 6,φ= real numb Ynm(0, )=spherical harmonic function Differential equation aY(6,φ) 1a2y(6,φ),1 +λY(6,中)=0 (n+1) Ym(6,中) 2n+1(n-m)! (E181) @2001 by CRC Press LLCIntegrals  Pn(x) dx = Pn+1(x) − Pn−1(x) 2n + 1 + C (E.170)  1 −1 xm Pn(x) dx = 0, m < n (E.171)  1 −1 x n Pn(x) dx = 2n+1(n!)2 (2n + 1)! (E.172)  1 −1 x 2k P2n(x) dx = 22n+1(2k)!(k + n)! (2k + 2n + 1)!(k − n)! (E.173)  1 −1 Pn(x) √1 − x dx = 2 √2 2n + 1 (E.174)  1 −1 P2n(x) √1 − x 2 dx =    n + 1 2  n! 2 (E.175)  1 0 P2n+1(x) dx = (−1) n (2n)! 2n + 2 1 (2nn!)2 (E.176) Fourier–Legendre series expansion of a function f (x) = ∞ n=0 an Pn(x), −1 ≤ x ≤ 1 (E.177) an = 2n + 1 2  1 −1 f (x)Pn(x) dx (E.178) E.3 Spherical harmonics Notation θ,φ = real numbers; m, n = integers Ynm(θ, φ) = spherical harmonic function Differential equation 1 sin θ ∂ ∂θ sin θ ∂Y (θ, φ) ∂θ + 1 sin2 θ ∂2Y (θ, φ) ∂φ2 + 1 a2 λY (θ, φ) = 0 (E.179) λ = a2 n(n + 1) (E.180) Ynm(θ, φ) =  2n + 1 4π (n − m)! (n + m)! Pm n (cos θ)e jmθ (E.181)