Page 582 - Handbook Of Integral Equations

Thus, by means of the Fourier transform, we succeeded in the passage from the original integral
equation to an algebraic equation for the transforms. However, in this case Eq. (19) involves two
unknown functions. In general, a single algebraic equation cannot uniquely determine two unknown
functions. The Wiener–Hopf method makes it possible to solve this problem for a certain class of
functions. This method is mainly related to the study of the analyticity domains of the functions that
enter the equation and to a special representation of this equation. The main idea of the Wiener–Hopf
method is as follows.
Let Eq. (19) be representable in the form

W + (z)Y + (z)= –W – (z)Y – (z), (21)

where the left-hand side is analytic in the upper half-plane Im z > µ and the right-hand side is
analytic in the lower half-plane Im z < v + , where µ < v + , so that there exists a common analyticity
strip of these functions: µ <Im z < v + . Since the analytic continuation is unique, it follows that there
exists a unique entire function of the complex variable that coincides with the left-hand side of (21)
in the upper half-plane and with the right-hand side of (21) in the lower half-plane, respectively.
If, in addition, the functions that enter Eq. (21) have at most power-law growth with respect to z
at infinity, then it follows from the generalized Liouville theorem (see Subsection 10.4-3) that the
entire function under consideration is a polynomial. In particular, for the case of a function that is
bounded at infinity we obtain
W + (z)Y + (z)= –W – (z)Y – (z) = const . (22)

These relations uniquely determine the functions Y + (z) and Y – (z).
Thus, let us apply the above scheme to the solution of Eq. (19). It follows from the above
√
reasoning that the analyticity domains of the functions Y + (z), Y – (z), and W(z)=1 – 2π K(z),
respectively, are the upper half-plane Im z > µ, the lower half-plane Im z < v + , and the strip
v – <Im z < v + . Therefore, this equation holds in the strip* µ <Im z < v + , which is the common
analyticity domain for all functions that enter the equation. In order to transform Eq. (19) to the
form (21), we assume that it is possible to decompose the function W(z) as follows:

W + (z)
W(z)= , (23)
W – (z)
where the functions W + (z) and W – (z) are analytic for Im z >µ and Im z <v + , respectively. Moreover,
we assume that, in the corresponding analyticity domains, these functions grow at infinity no faster
n
than z , where n is a positive integer. A representation of an analytic function W(z) in the form (23)
is often called a factorization of W(z).
Thus, as the result of factorization, the original equation is reduced to the form (21). It follows
from the above reasoning that this equation determines an entire function of the complex variable z.
Since Y ± (z) → 0as |z| →∞ and the growth of the functions W ± (z) does not exceed that
n
of a power function z , it follows that the entire function under consideration can be only a
polynomial P n–1 (z) of degree at most n – 1.
If the growth of the functions W ± (z)atinfinity is only linear with respect to the variable z, then
it follows from relations (22), by virtue of the Liouville theorem (see Subsection 10.4-3), that the
corresponding entire function is a constant C. In this case we obtain the following relations for the
unknown functions Y + (z) and Y – (z):

C C
Y + (z)= , Y – (z)= – , (24)
W + (z) W – (z)
* Tobedefinite, we set µ > v –. Otherwise, the common domain of analyticity is the strip v – <Im z < v +.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 565