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are reduced to
∫
∫
x'
x
x'
−x
Σ(
t)
dt = Σ
Σ(
t)
dt = Σ
( x
− x)
+ Σ
( x − x
= Σ ( x − x
where λ Σ x − x ) .
k = k ( k k −1
i
i
i
i
i−1
i−1
j
) + Σ
) + Σ
( x'
−x
j
j
( x'
−x
( x'
−x
j−1
j−1
) +
j−1
) +
) +
j−1
∑
k
k = i+
1
j−1
∑
k = 1
λ
j−1
∑
k
k = i+
1
,
λ + 2
λ +
Then we can carry out the integration over x and x’, and we get
P
ij
2
=
Σ V
i
∫
r
0
⎪⎧
⎛
× ⎨ exp⎜
−
⎪⎩ ⎝
i
i
/ 2
ρ∫
π
0
d
i−1
∑
k = 1
λ / sinθ
−
k
⎞⎪⎫
λ / sin ⎟
k θ
⎬ .
⎠⎪⎭
k
i−1
∑
k = 1
i−1
∑
k = 1
λ ,
k
λ + λ
⎧ ⎛ λ ⎞⎫⎪⎧
⎛ λ ⎞⎪⎫
sin ⎨1
− exp⎜
−
i
j
θdθ
⎟⎬⎨1
− exp⎜
⎟⎬
sin
−
⎪⎩ sin
⎩ ⎝ θ ⎠⎭
⎝ θ ⎠⎪⎭
j−1
∑
k = 1
Now we introduce the Bickley-Naylor function 50)
π / 2
⎛ ⎞
=
n−1
x
K
in ( x)
∫ dθ
sin θ exp⎜
− ⎟ (7.1-45)
0
⎝ sinθ
⎠
We have the final form of P ij for the case r i