Page 539 - Electromagnetics

n 1
+ nπ
2 2
P n (0) = √ n cos (E.155)
π + 1 2
2
(n − m)! m
m
−m
P (x) = (−1) P (x) (E.156)
n n
(n + m)!
Power series
n k

(−1) (n + k)! k n k
P n (x) = 2 k+1 (1 − x) + (−1) (1 + x) (E.157)
(n − k)!(k!) 2
k=0
Recursion relationships
m
m
(n + 1 − m)R n+1 (x) + (n + m)R m (x) = (2n + 1)xR (x) (E.158)
n−1
n
2
m
m
(1 − x )R (x) = (n + 1)xR (x) − (n − m + 1)R m (x) (E.159)
n n n+1
(2n + 1)xR n (x) = (n + 1)R n+1 (x) + nR n−1 (x) (E.160)
2

(x − 1)R (x) = (n + 1)[R n+1 (x) − xR n (x)] (E.161)
n
R (x) − R (x) = (2n + 1)R n (x) (E.162)
n+1 n−1
Integral representations

√
2 π sin n + 1 2 u
P n (cos θ) = √ du (E.163)
π 0 cos θ − cos u
1 π 2 1/2 n
P n (x) = x + (x − 1) cos θ dθ (E.164)
π 0
Addition formula

P n (cos γ) = P n (cos θ)P n (cos θ ) +
n
(n − m)! m m

+ 2 P (cos θ)P (cos θ ) cos m(φ − φ ), (E.165)
n
n
(n + m)!
m=1

cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.166)
Summations
∞
1 1 r n <
= P n (cos γ) (E.167)
= n+1
2 2

|r − r | r + r − 2rr cos γ n=0 r >

cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.168)

r < = min |r|, |r | , r > = max |r|, |r | (E.169)
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