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where the integration is performed over any line Im z = v > v – in the complex plane z, which is
parallel to the real axis.
For v – < 0 (i.e., for functions y(x) with exponential decay at infinity), the real axis belongs
to the domain in which the function Y + (z) is analytic, and we can integrate over the real axis in
+
formula (6). However, if the only possible values of v – are positive (for instance, if the function y (x)
has nontrivial growth at infinity, which does not exceed the exponential growth with linear exponent),
then the analyticity domain of the function Y + (z) is strictly above the real axis of the complex plane z
–
(and in this case, the integral (4) can be divergent on the real axis). Similarly, if the function y (x)
in relations (3) satisfies the condition
–
|y (x)| < Me v + x as x → –∞, (7)

then its transform, i.e., the function
0
1 – izx
Y – (z)= √ y (x)e dx, (8)
2π –∞
–
is an analytic function of the complex variable z in the domain Im z < v + . The function y (x) can
be expressed via Y – (z) by means of the relation

1 ∞+iv
– –izx
y (x)= √ Y – (z)e dz, Im z = v < v + . (9)
2π –∞+iv
For v + > 0, the analyticity domain of the function Y – (z) contains the real axis.
It is clear that for v – < v + , the function Y(z)defined by formula (1) is an analytic function of the
complex variable z in the strip v – <Im z < v + . In this case, the functions y(x) and Y(z) are related
by the Fourier inversion formula

∞+iv
1 –izx
y(x)= √ Y(z)e dz, (10)
2π –∞+iv
where the integration is performed over an arbitrary line in the complex plane z belonging to the
strip v – <Im z < v + . In particular, for v – < 0 and v + > 0, the function Y(z) is analytic in the strip
containing the real axis of the complex plane z.
Example 1. For α > 0, the function K(x)= e –α|x| has the transform
1 2α
K(z)= √ ,
2
2π α + z 2
which is an analytic function of the complex variable z in the strip –α <Im z < α, which contains the real axis.

11.10-2. The Homogeneous Wiener–Hopf Equation of the Second Kind
Consider a homogeneous integral Wiener–Hopf equation of the second kind in the form

∞

y(x)= K(x – t)y(t) dt, (11)
0
whose solution can obviously be determined up to an arbitrary constant factor only. Here the domain
of the function K(x) is the entire real axis. This factor can be found from additional conditions of
the problem, for instance, from normalization conditions.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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