Page 540 - Electromagnetics

Integrals
P n+1 (x) − P n−1 (x)

P n (x) dx = + C (E.170)
2n + 1
1
m
x P n (x) dx = 0, m < n (E.171)
−1
1 2 n+1 (n!) 2
n
x P n (x) dx = (E.172)
−1 (2n + 1)!
1 2 2n+1 (2k)!(k + n)!
2k
x P 2n (x) dx = (E.173)
−1 (2k + 2n + 1)!(k − n)!
√
1
P n (x) 2 2

√ dx = (E.174)
−1 1 − x 2n + 1
1 1 2
P 2n (x) n + 2

√ dx = (E.175)
−1 1 − x 2 n!
1
(2n)! 1
P 2n+1 (x) dx = (−1) n (E.176)
n
0 2n + 2 (2 n!) 2
Fourier–Legendre series expansion of a function
∞

f (x) = a n P n (x), −1 ≤ x ≤ 1 (E.177)
n=0
2n + 1 1
a n = f (x)P n (x) dx (E.178)
2 −1

E.3 Spherical harmonics
Notation

θ, φ = real numbers; m, n = integers
Y nm (θ, φ) = spherical harmonic function

Differential equation
2
1 ∂ ∂Y(θ, φ) 1 ∂ Y(θ, φ) 1
sin θ + + λY(θ, φ) = 0 (E.179)
2
sin θ ∂θ ∂θ sin θ ∂φ 2 a 2
2
λ = a n(n + 1) (E.180)

2n + 1 (n − m)! m
Y nm (θ, φ) = P (cos θ)e jmθ (E.181)
n
4π (n + m)!

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