Page 537 - Electromagnetics

m
Q (cos θ) = associated Legendre function of the second kind
n
0
P n (cos θ) = P (cos θ) = Legendre polynomial
n
0
Q n (cos θ) = Q (cos θ) = Legendre function of the second kind
n
Differential equation x = cos θ.
2
m
m
d R (x) dR (x) m 2
m
n
n
2
(1 − x ) − 2x + n(n + 1) − R (x) = 0, −1 ≤ x ≤ 1 (E.116)
n
dx 2 dx 1 − x 2
m
P (x)
m n
R (x) = (E.117)
m
n Q (x)
n
Orthogonality relationships
1
m m 2 (n + m)!
P (x)P (x) dx = δ ln (E.118)
n
l
−1 2n + 1 (n − m)!
π 2 (n + m)!
m m
P (cos θ)P (cos θ) sin θ dθ = δ ln (E.119)
l n
0 2n + 1 (n − m)!
P (x)P (x) 1 (n + m)!
1 m k
n
n
dx = δ mk (E.120)
1 − x 2 m (n − m)!
−1
π m k
P (cos θ)P (cos θ) 1 (n + m)!
n n
dθ = δ mk (E.121)
0 sin θ m (n − m)!
2
1
P l (x)P n (x) dx = δ ln (E.122)
−1 2n + 1
2
π
P l (cos θ)P n (cos θ) sin θ dθ = δ ln (E.123)
0 2n + 1
Specific examples
P 0 (x) = 1 (E.124)
P 1 (x) = x = cos(θ) (E.125)
1 2 1
P 2 (x) = (3x − 1) = (3 cos 2θ + 1) (E.126)
2 4
1 3 1
P 3 (x) = (5x − 3x) = (5 cos 3θ + 3 cos θ) (E.127)
2 8
1 4 2 1
P 4 (x) = (35x − 30x + 3) = (35 cos 4θ + 20 cos 2θ + 9) (E.128)
8 64
1 5 3 1
P 5 (x) = (63x − 70x + 15x) = (63 cos 5θ + 35 cos 3θ + 30 cos θ) (E.129)
8 128
1 1 + x θ
Q 0 (x) = ln = ln cot (E.130)
2 1 − x 2
x 1 + x θ
Q 1 (x) = ln − 1 = cos θ ln cot − 1 (E.131)
2 1 − x 2
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