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⎪⎧
⎪⎧
⎪⎫
j−1
1 π / 2
⎛ λ ⎞⎫
⎛ λ ⎞ ⎧ ⎫
= ⎨ − ⎜ −
i
j
λ
∫
⎟⎬
⋅ ⎨ − ⎜ ⎟
−
⎬ ⋅ ⎨ −
k
P ij sinθ
cosθ
dθ
1 exp
1 exp
exp ∑ ⎬
2λ
0
i
⎪⎩ ⎝ cosθ
⎠⎭
⎪⎩ ⎝ cosθ
⎠⎪⎭
⎩ k = i+
1 cosθ
⎭
Now we introduce the exponential integral function, E in defined by Schloemich 49)
E
in
( x)
= ∫
1
0
dμ
μ
n−1
⎛ x ⎞
exp⎜
− ⎟
⎝ μ ⎠
We have the final form of P ij for the case x i < x j as follows:
P
ij
{ E ( λ ) − E ( λ + λ ) − E ( λ + λ ) + E ( λ + λ + λ )}
1
= i3 ij i3
ij i i3
ij j i3
ij i j
(7.1-26)
2λ
i
j
∑ − 1
k = i+
1
where λ ij = λ k
, for x i x j , the optical distance is reduced to
x'
i 1
∫ ∑ − Σ(
t)
dt = Σ i ( x − xi− 1 ) + Σ j ( x j − x'
) +
x
k = j+
by using the same procedure as x i x j (7.1-28)
In the last case where x i = x j , the optical distance in Eq.(7.1-25) is reduced to
∫
x
x'
Σ(
t)
dt
=
⎧Σ
⎨
⎩Σ
i
i
( x'
−x)
for x'
> x
( x − x'
) for x > x'
Integrating over x and x’ gives the final form of P ii by
1
Pii
= 1−
{ Ei3(0)
− Ei3(
λi
)}
(7.1-29)
λ
i
If the λ i ’s are so small that the differences in Eq.(7.1-26) and in Eq.(7.1-29) can not be obtained
accurately in the numerical calculation, we should use the following differential forms instead of
Eqs.(7.1-26) and (7.1-29), respectively,
P
P
ij
ii
λ j
= Ei1(
λij
+ λi
/ 2 + λ j / 2) ,
(7.1-30)
2
= λ E ( λ / 2)
(7.1-31)
i
i1 i
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