6.3 Explicit Expressions for the Integrals with Vector …
111
I
21
mnm n = (−1)
(m+m
)
π
0
n
(n
+ 1)
x 1
b
n
om (θ )d
n
om (θ ) sin(θ )
2π
0
f
3
mnm n (θ, φ)dϕ
−
−im
n
(n
+ 1)
x 1
d
n
om (θ )d
n
om (θ )/ sin(θ )
2π
0
f
4
mnm n (θ, φ)dϕ
+
+ (mm
d
n
om (θ )d
n
om (θ )/sin(θ )+
(6.31)
+b
n
om (θ )b
n
om (θ )sin(θ )
2π
0
f
5
mnm n (θ, φ)dϕ
dθ,
I
22
mnm n = (−1)
(m+m )
π
0
i(md
n
om (θ)b
n
om (θ) + m
d
n
om (θ)b
n
om (θ))
2π
0
f
6
mnm n (θ, φ)dϕ
+
+
n (n + 1)
x 1
b
n
om (θ)d
n
om (θ)
2π
0
f
7
mnm n (θ, φ)dϕ
−
n(n + 1)
x
d
n
om (θ)b
n
om (θ)×
×
2π
0
f
8
mnm n (θ, φ)dϕ
+ im
n (n + 1)
x 1
d
n
om (θ)d
n
om (θ)
2π
0
f
9
mnm n (θ, φ)dϕ
+
+ im
n(n + 1)
x
d
n
om (θ )d
n
om (θ )
2π
0
f
10
mnm n (θ, φ)dϕ
dθ,
(6.32)
where
c
1
mnm n (θ, φ) = exp[i m m ]h n (x) j n (x 1 )r
2
(θ, φ),
c
2
mnm n (θ, φ) = exp[i m m ]u n (x) j n (x 1 )r
2
(θ, φ),
c
3
mnm n (θ, φ) = exp[i m m ]h n (x) j n (x 1 )
dr(θ, φ)
dθ
,
c
4
mnm n (θ, φ) = exp[i m m ]h n (x) j n (x 1 )
dr(θ, φ)
dθ
,
c
5
mnm n (θ, φ) = exp[i m m ]h n (x)v n (x 1 )r
2
(θ, φ),
c
6
mnm n (θ, φ) = exp[i m m ]u n (x)v n (x 1 )r
2
(θ, φ),
c
7
mnm n (θ, φ) = exp[i m m ]u n (x) j n (x 1 )
dr(θ, φ)
dϕ
,
c
8
mnm n (θ, φ) = exp[i m m ]h n (x)v n (x 1 )
dr(θ, φ)
dϕ
,