Page 542 - Electromagnetics

cos γ = cos θ cos θ + sin θ sin θ cos(φ − φ ) (E.198)

Series
n
2n + 1
2
|Y nm (θ, φ)| = (E.199)
4π
m=−n
1 1
=
2
2

|r − r | r + r − 2rr cos γ
∞ n n
1 r <

∗

= 4π Y (θ ,φ )Y nm (θ, φ), (E.200)
n+1 nm
n=0 m=−n 2n + 1 r >

r < = min |r|, |r | , r > = max |r|, |r | (E.201)
Series expansion of a function
∞ n

f (θ, φ) = a nm Y nm (θ, φ) (E.202)
n=0 m=−n
π π
∗
a nm = f (θ, φ)Y (θ, φ) sin θ dθ dφ (E.203)
nm
−π 0

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