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where the function G + (z) is analytic in the half-plane Im z > v – , the function G – (z) is analytic in the
half-plane Im z < v + , and

1 ∞+iv – G(τ)
G + (z)= dτ, v – < v <Im z < v + , (41)

–
2πi –∞+iv τ – z
–

1 ∞+iv + G(τ)
G – (z)= – dτ, v – <Im z < v < v + . (42)

+
2πi τ – z
–∞+iv +
The integrals (41) and (42), being regarded as integrals depending on a parameter, define analytic
functions of the complex variable z under the assumption that the point z does not belong to the
integration contour.
In particular, G + (z) is an analytic function in the half-plane Im z > v and G – (z) in the half-plane

–

Im z > v .
+
Moreover, if a function H(z) is analytic and nonzero in the strip v – <Im z < v + and if H(z) → 1
uniformly in this strip as |z| →∞, then the following representation holds in the strip:
H(z)= H + (z)H – (z), (43)

1 ∞+iv – ln H(τ)
H + (z)=exp dτ , v – < v <Im z < v + , (44)

–
2πi –∞+iv τ – z
–

1 ∞+iv + ln H(τ)

H – (z)=exp – dτ , v – <Im z < v < v + , (45)
+
2πi τ – z
–∞+iv +
where the functions H + (z) and H – (z) are analytic and nonzero in the half-planes Im z > v – and
Im z < v + , respectively. The representation (43) is called a factorization of the function H(z).
11.10-4. The Nonhomogeneous Wiener–Hopf Equation of the Second Kind
Consider the Wiener–Hopf equation of the second kind

∞
y(x) – K(x – t)y(t) dt = f(x), (46)
0
Suppose that the kernel K(x) of the equation and the right-hand side f(x) satisfy conditions (15).
+
Let us seek the solution y (x) to Eq. (46) for which condition (17) is satisfied.
In this case, reasoning similar to that in the derivation of the functional equation (19) for a
homogeneous integral equation shows that, in the case of Eq. (46), the following functional equation
must hold on the strip µ <Im z < v + :
√
Y + (z)+ Y – (z)= 2π K(z)Y + (z)+ F + (z)+ F – (z), (47)
or
W(z)Y + (z)+ Y – (z) – F(z) = 0, (48)
where W(z) is subjected to condition (20), as well as in the case of a homogeneous equation.
We now note that Eq. (48) is a special case of Eq. (34). In the strip v – <Im z < v + , the
function W(z) is analytic and uniformly tends to 1 as |z| →∞ because |K(z)| → 0as |z| →∞.In
this case, this function has the representation (see (43)–(45))

W + (z)
W(z)= , (49)
W – (z)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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