Page 541 - Electromagnetics

Orthogonality relationships
π π
∗
Y (θ, φ)Y nm (θ, φ) sin θ dθ dφ = δ n n δ m m (E.182)

n m
−π 0
∞ n

∗

Y (θ ,φ )Y nm (θ, φ) = δ(φ − φ )δ(cos θ − cos θ ) (E.183)
nm
n=0 m=−n
Specific examples

1
Y 00 (θ, φ) = (E.184)
4π

3
Y 10 (θ, φ) = cos θ (E.185)
4π

3 jφ
Y 11 (θ, φ) =− sin θe (E.186)
8π

5 3 2 1
Y 20 (θ, φ) = cos θ − (E.187)
4π 2 2

15 jφ
Y 21 (θ, φ) =− sin θ cos θe (E.188)
8π

15 2 2 jφ
Y 22 (θ, φ) = sin θe (E.189)
32π

7 5 3 3
Y 30 (θ, φ) = cos θ − cos θ (E.190)
4π 2 2

21 2 jφ
Y 31 (θ, φ) =− sin θ 5 cos θ − 1 e (E.191)
64π

105 2 2 jφ
Y 32 (θ, φ) = sin θ cos θe (E.192)
32π

35 3 3 jφ
Y 33 (θ, φ) =− sin θe (E.193)
64π
Functional relationships

2n + 1
Y n0 (θ, φ) = P n (cos θ) (E.194)
4π
m
Y n,−m (θ, φ) = (−1) Y (θ, φ) (E.195)
∗
nm
Addition formulas
n
4π

∗

P n (cos γ) = Y nm (θ, φ)Y (θ ,φ ) (E.196)
nm
2n + 1
m=−n

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

+ P (cos θ)P (cos θ ) cos m(φ − φ ) (E.197)
n
n
(n + m)!
m=−n
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