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earrajando os termos,
2 π
∗
∗ πk
∗ ∗ ∗ [ ρ h ′ ρ ′
∫ θ ϕ ϕ=− ∑
2
n ( )] m m m
Es Hs d αnmα n′ m{[ i aenmaen′ m + aonmaon′ m] hn( ρ)
P
2 n Pn′
µω , ′,
ρ sen θ
0
nn m
m m
∗ ∗ dPn dPn′
[ ρ hn( ρ)]
′ ∗
− ib [ enmben′ m(1 +δ m0) + bonmbon′ m(1 −δm0)] hn′
( ρ)
dθ dθ ρ
m m m m
∗
∗
∗ mPn
dPn′ ∗ ∗ ∗ mPn dPn′ [ ρ hn( ρ)] ′ [ ρ hn′
( ρ)]
′
+− [ aenmbon′ m + aonmben′
m] hn( ρ) hn′ ( ρ ) + [ benmaon′ m −bonmaen′
m] }
sen θ dθ sen θ dθ ρ ρ
(3. 79)
e
2π
∫
0
m m
′
E H d i a a a a h
∗
∗ πk
∗ ∗ ∗ [ ρ hn′ ( ρ)]
dPn dPn′
sϕ
sθ
ϕ=− ∑ αnmαnm ′ { − [ enm enm ′ (1 +δ m0) + onm onm ′ (1 −δm0)] n( ρ)
µω nn , ′,
m
ρ dθ dθ
2
∗ ∗ m m m [ ρ hn( ρ)]
′ ∗
+ ib [ enmben′ m + bonmbon′ m] P ′ ′( ρ)
2 n Pn hn
sen θ ρ
m m m m
∗
∗
∗ mPn
dPn′ ∗ ∗ ∗ mPn dPn′ [ ρ hn( ρ)] ′ [ ρ hn′
( ρ)]
′
+− [ aenmbon′ m −aonmben′
m] hn( ρ) hn′ ( ρ ) + [ − benmaon′ m + bonmaen′
m] }
sen θ dθ sen θ dθ ρ ρ
(3. 80)
tal que no final,
π 2π
∫∫
0 0
( E H −E H ) sen θ dθ dϕ
∗
∗
sθ sϕ sϕ sθ
∗ π m m 2
∗ ∗ ∗ [ ρ hn′ ( ρ)]
′ dPn dPn′
m m m
∑ nm n′ m enm en′ m m0 onm on′ m m0 n ∫
2 n n′
nn , ′, m
d d sin
0
πk
=− α α {[ ia a (1 +δ ) + a a (1 −δ )] h( ρ ) ( + P P ) sen θ dθ
µω ρ θ θ θ
m m 2
∗
∗ [ ρ hn( ρ)]
′ ∗ dPn dPn′
m m m
− ib [ enmben′ m(1 +δ m0) + bonmbon′
m(1 −δm0)]
hn′ ( ρ ) + ′ θ θ
ρ
∫( P ) sen
2 n Pn
d
dθ dθ sen θ
π
0
π m m m m
∗ ∗ ∗ mPn dPn′ mPn′
dPn
enm onm ′ onm enm ′ n n′
∫
sen θ dθ sen θ dθ
0
+− [ a b + a b ] h ( ρ) h ( ρ ) ( + ) sen θdθ
(3. 81)
∗ π m m m m
∗ ∗ [ ρ hn( ρ)] ′[ ρ hn′ ( ρ)]
′ mPn dPn′ mPn′
dPn
enm on′ m onm en′
m
ρ ρ
∫
sen θ dθ sen θ dθ
0
+ [ b a − b a ] ( + ) senθdθ
As integrais em θ são de dois tipos:
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