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or
ik
coth
k
+
τ
K
2
=
τ
Im
k
coth
k
2
+Re
2
k
coth
k
−
Re
2
k
coth
k
≥
(
R
cot
R
)
2
,
≥
where
k
<R<π/
2
.
The variable
k
2
<R
2
+
k
A
in the circle of radius
k
A
−
R.
Therefore (8.38) has no roots for
R
2
+
k
A
<R
cot
R.
Since
k
A
1, the
last inequality is true within the circle of radius
R
0
≈
π/
4
.
Hence (8.38) has
no roots.
Let
k
>π/
4
.
Expand
k
A
−
k
2
1
/
2
into power series of
k
A
/k
:
ik
1
2
k
2
+
...
k
A
k
2
k
A
k
A
−
k
2
1
/
2
=
ik
±
1
−
=
±
−
The parameter
k
A
∝
T
−
1
and for
T>
10 s, the value of
k
A
<
10
−
1
.
We leave
the first term in the expansion of the root, then
k
A
−
k
2
1
/
2
ik.
≈±
The sign here is chosen according to the following rule: plus is on the physical
sheet in the first quadrant and minus is on the nonphysical sheet and vice
versa. Therefore (8.38) reduces in the 1-st quadrant on the physical sheet and
in the 2-nd quadrant of the nonphysical sheet to
2
ik
exp
−
2
k
=
−
τ
K
.
(8.A.2)
1
−
In the 2-nd quadrant on the physical sheet and in the 1-st quadrant of the
nonphysical one, we have
2
ik
exp
2
k
=
−
−
τ
K
.
(8.A.3)
1
−
−
Equations (8.A.2) and (8.A.3) are tantamount to an equations set for
x
=Re
k
and
y
=Im
k
:
Phys.
sheet (
I
)
=
⇒
2
y
=
τ
K
(1
−
exp (
−
2
x
)cos2
y
)
Nonphys. sheet (
II
)
=
⇒
2
x
=
−
τ
K
exp (
−
2
x
)sin2
y,
(8.A.4)
and
Phys. sheet (
II
)
=
⇒−
2
y
=
τ
K
(1
−
exp (2
x
)cos2
y
)
Nonphys. sheet (
I
)
=
⇒
2
x
=
−
τ
K
exp (2
x
)sin2
y.
(8.A.5)
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