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where the function W + (z) is analytic in the upper half-plane Im z > v – and W – (z) is analytic in the
lower half-plane Im z < v + , and the growth at infinity of the functions W ± (z) does not exceed that
n
of z .
On the basis of the representation (49), Eq. (48) becomes

W + (z)Y + (z)+ W – (z)Y – (z) – W – (z)F – (z) – W – (z)F + (z) = 0. (50)
To reduce Eq. (50) to the form (38), it suffices to decompose the last summand

F + (z)W – (z)= D + (z)+ D – (z) (51)

into the sum of functions D + (z) and D – (z) that are analytic in the half-planes Im z > µ and Im z < v + ,
respectively.
To establish the possibility of a representation (51), we note that the function F + (z) is analytic
in the upper half-plane Im z > v – and uniformly tends to zero as |z| →∞. The function W – (z)is
analytic in the lower half-plane Im z < v + , and, according to the method of its construction, we can
perform the factorization (49) so that the function W – (z) remains bounded in the strip v – <Im z < v +
as |z| →∞. Hence (see (40)–(42)), the functions F + (z)W – (z) in the strip v – <Im z < v + satisfy all
conditions that are sufficient for the validity of the representation (51).
The above reasoning makes it possible to take into account the fact that the growth at infinity of
n
the functions W ± (z) does not exceed that of z , and thus to present the transform of the solution of
the nonhomogeneous integral equation (46) in the form
P n–1 (z)+ D + (z) –P n–1 (z)+ W – (z)F – (z)+ D – (z)
Y + (z)= , Y – (z)= . (52)
W + (z) W – (z)
The solution itself can be obtained from (52) by means of the Fourier inversion formula (6), (9),
and (10).

11.10-5. The Exceptional Case of a Wiener–Hopf Equation of the Second Kind
Consider the exceptional case of a Wiener–Hopf equation of the second kind in which the func-
√
tion W(z)=1 – 2π K(z) has finitely many zeros N (counted according to their multiplicities) in
the strip v – <Im z < v + . In this case, the factorization is also possible. To this end, it suffices to
introduce the auxiliary function

2 N/2
2
W 1 (z)=ln (z + b ) W(z) (z – z i ) –α i , (53)
i
where α i is the multiplicity of the zero z i and a positive constant b > {|v – |, |v + |} is chosen so that the
function in the square brackets has no additional zeros in the strip v – <Im z < v + .
However, in the exceptional case, the Wiener–Hopf method gives the answer only if the number
of zeros of the function W(z) is even. This restriction is due to the fact that only for the case in
which the number of zeros is even is it possible to achieve the necessary behavior at infinity (for
2 N/2
2
the application of the Wiener–Hopf method) of the function (z + b ) (see F. D. Gakhov and
Yu. I. Cherskii (1978)). The last restriction makes no real obstacle to the broad use of the Wiener–
Hopf method in solving applied problems in which the kernel K(x) of the corresponding integral
equation is frequently an even function, and thus the reasoning below can be applied completely.
Remark 1. The Wiener–Hopf equation of the second kind for functions vanishing at infinity can
be reduced to a Riemann boundary value problem on the real axis (see Subsection 11.9-1). In this
case, the assumption that the number of zeros of the function W(z) is even, as well as the assumption
that the kernel K(x) is even in the exceptional case, are unessential.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 569