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m′ m′
∗ −k
∗ ∗ dPn′ ∗ ∗ dPn′
∗
sϕ
∑ nm ′ ′ enm ′ ′ ′
n′ onm ′ ′ ′
n′
µω
n′ , m′
dθ dθ
H = α { b cos mϕ h ( ρ) −b sen mϕ h ( ρ)
m′ ∗
m′
∗ mP ′
n′ [ ρ hn′ ( ρ)] ′ ∗ mP ′
n′ [ ρ hn′
( ρ)]
′
+ iaen′ m′ sen m′ ϕ + iaon′ m′
cos m′
ϕ }
sen θ ρ sen θ ρ
∗
(3. 74)
m
m
mPn
mPn
Esϕ = ∑ α nm{ − aenm cos m hn( ) aonm sen m hn( )
nm
sen θ
ϕ ρ − sen θ
ϕ ρ
(3. 75)
m
m
dPn [ ρ hn( ρ)] ′ dPn [ ρ hn( ρ)]
′
+ ibenm
sen mϕ −ibonm
cos mϕ
}
dθ ρ dθ ρ
m′ m′
∗ −k
∗ ∗ dPn′ ∗ ∗ dPn′
∗
Hsθ
= ∑ αnm ′ ′{ benm ′ ′ sen m′ ϕ hn′ ( ρ) −bonm ′ ′ cos m′
ϕ hn′
( ρ)
µω dθ dθ
nm ′ ′
′ ′
m
∗
m
∗
∗ mP ′
n′ [ ρ hn′ ( ρ)] ′ ∗ mP ′
n′ [ ρ hn′
( ρ)]
′
+ iaen′ m′ cos m′ ϕ + iaon′ m′
sen m′
ϕ }
sen θ ρ sen θ ρ
(3. 76).
2π
∫
0
temos,
Agora, podemos integrar em ϕ e utilizar o fato de que:
cos mϕ cos m′
ϕ d ϕ= (1 +δ ) πδ ,
2π
∫
0
m0
mm′
2π
2π
0 0
2π
∫ sen mϕsen m′
ϕ dϕ= (1 −δm0
) πδmm′
e
0
∫ cos mϕ sen m′ ϕ dϕ= 0 = ∫ sen mϕ cos m′
ϕ dϕ
(3. 77)
m m
πk
mP dP
E H dϕ=− α α { −a b (1 −δ ) h ( ρ) h ( ρ ) +
∑
∗ ∗ ∗ n n′
∗
sθ
sϕ
nm nm ′ enm onm ′ m0
n n′
µω
nn , ′,
m
sen θ dθ
2
∗
∗ m m m ρ hn′
enm en′ m(1 m0) 2 n n′
n( )
[ ( ρ)]
′
ia a −δ P P h ρ +
sen θ
ρ
m m
∗ n n′
∗ ∗
onm en′ m +δm0 n ρ n′ ρ + onm on′
m +δm0
mP dP m
a b (1 ) h ( ) h ( ) ia a (1 ) P P h
2
sen θ dθ
sen θ
2
∗
m m ρ hn
′
n n′
n( ρ)
m m m m
∗
∗ dPn dPn′ hn ∗ ∗
mPn dPn′ hn hn′
enm enm ′ m0 n′ enm onm ′ m0
m m
∗ dPn dPn′
[ ρ hn( ρ)]
′ ∗ ∗
onm on′ m m0
n′ onm en′
m
[ ( ρ)]
′
ρ
[ ρ ( ρ)] ′ [ ρ ( ρ)] ′[ ρ ( ρ)]
′
− ib b (1 +δ ) h ( ρ ) + b a (1 +δ )
dθ dθ ρ sen θ dθ ρ ρ
−ib b (1 −δ ) h ( ρ) −b a (1−δ
dθ dθ ρ
(3. 78)
m0
m m
∗
n n′ n n′
mP dP [ ρ h ( ρ)] ′ [ ρ h ( ρ)]
′
) }
sen θ dθ ρ ρ
91