The Wiener-Hopf method is a powerful technique which enables certain linear partial differential equations subject to boundary conditions on semi-infinite domains to be solved explicitly. The method is sometimes referred to as the Wiener-Hopf technique or the Wiener-Hopf factorization.
The Wiener-Hopf method begins by applying the generalized upper and lower Fourier transforms to obtain an identity
 |
(1)
|
on a strip
 |
(2)
|
of the complex plane with coordinate 
where 
is a complex
variable. Note that identity (1)
is in terms of the unknown functions 
and 
which are analytic
in the half-planes 
and 
, respectively, while 
, 
, and 
are "parameter functions" in the complex
plane which are analytic on all of 
.
For simplicity, assume that 
and 
are nonzero in 
. The most fundamental step of the Wiener-Hopf process is to
find a solution for 
and 
in (1) by finding functions 
and 
--analytic and
nonzero in 
and in 
, respectively--so that
 |
(3)
|
Upon doing so, the factorization (3) can be used to rewrite (1) as
 |
(4)
|
whereby the last addend 
can be decomposed as
 |
(5)
|
for 
,
respectively 
,
analytic throughout the half-planes 
, respectively 
.
Substituting (5) into (4) and rewriting induces a function 
of the form
 |
(6)
|
whose first expression is analytic throughout 
and whose second expression is analytic throughout
.
Since they agree on the overlap 
, the two expressions define one analytic function on the entire complex
plane. The idea behind (6) is to next show the existence
of positive integers 
for which
 |
(7)
|
and
 |
(8)
|
whereby Liouville's theorem applies and requires that 
be a polynomial 
of degree less
than or equal to 
.
In particular,
 |
(9)
|
and
 |
(10)
|
thus defining 
and 
to within the arbitrary polynomial 
, i.e., to within a finite number of arbitrary constants
which must be determined using other methods.
While the Wiener-Hopf method itself is a useful tool for solving various types of partial differential equations, one of its most significant strengths is the vast array of other equation solving methods derived therefrom. Indeed, the techniques spawned from the Wiener-Hopf factorization have proven useful in a number of very different circumstances across a diverse array of disciplines including theoretical and applied physics (Noble 1958), diffraction theory (Linton and McIver 2001), and fluid dynamics (Ho 2007).
