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Im
k
Im
k
g
= 0
k
S
= 0
Im
e
g
1/2
k
g
= 0
k
0
Im
Re
k
S
= 0
Re
k
g
= 0
Γ
1
k
0
e
m
1/2
−
k
0
e
m
1/2
Re
k
k
S
= 0
Re
Re
k
g
= 0
Im
k
S
= 0
−
k
0
e
g
1/2
Im
k
g
= 0
Fig. 8.1.
Integration path
Γ
1
going alon
g the real
axis over
the physica
l sheet of the
complex wave number
k
plane.
κ
g
=
k
0
ε
g
− k
2
,κ
s
=
k
0
ε
m
− k
2
,ε
g
=2
iσ
g
T
.
Four points of branching
g
and
± k
0
ε
1
/
m
are denoted by solid dots. Solid
lines are the cut lines Im
κ
s
= 0 and Im
κ
g
=0
±k
0
ε
1
/
2
∓
|
|
√
ε
m
b
(
i,r
)
sin
I
E
(
i,r
)
x
E
(
i,r
)
z
E
(
i,r
)
x
(
k
)=
(
k
)
,
(
k
)=
−
(
k
)cot
I,
(8.3)
y
k
0
k
κ
S
b
(
i,r
)
E
(
i,r
)
y
κ
S
b
(
i,r
)
b
(
i,r
)
z
(
k
)=
±
(
k
)
,
(
k
)=
±
(
k
)
,
(8.4)
x
x
c
4
π
b
(
i,r
)
(
k
)
±
k
A
+
k
cos
I
sin
I
c
4
π
kb
(
i,r
)
j
(
i,r
)
x
(
i,r
)
z
(
k
)=
i
,
(
k
)=
i
(
k
)
,
(8.5)
y
y
with
κ
S
=(
k
A
−
k
2
)
1
/
2
,
k
A
=
ω/c
A
.
The upper and lower signs refer here, re-
spectively, to the incident and reflected waves. Their coordinate dependencies
may be presented as
1
√
2
π
R
(
k
)
b
(
i
τ
(
k
)exp(
ikx
)
dk,
b
(
r
)
τ
(
x,
+0) =
Γ
1
1
√
2
π
b
(
g
τ
(
x
)=
b
τ
(
x,
T
(
k
)
b
(
i
τ
(
k
)exp(
ikx
)
dk.
−
h
)=
(8.6)
Γ
1
along
the real axis over the physical sheet of the complex plane
k
where Im
κ
S
>
0
,
Im
κ
g
>
0 (see Table 7.2).
Figure 8.1 shows the integration contour
Γ
1
running from
−∞
to +
∞
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