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.

of the complex -plane where is a complex
variable. Note that identity () is in terms of the unknown functions and which are analytic
in the half-planes and , respectively, while , , and are "parameter functions" in the -plane which are analytic on all of .

For simplicity, assume that and are non-zero in . The most fundamental step of the Wiener-Hopf process is to
find a solution for
and
in () by finding functions and --analytic and nonzero
in
and in ,
respectively--so that

(3)

Upon doing so, the factorization () can be used to rewrite () as

for ,
respectively ,
analytic in the region of satisfying , respectively .

Substituting () into () and rewriting induces a function of the form

(6)

which, despite being defined only in the strip , can be defined and made analytic on the entire complex -plane by way of analytic
continuation. The idea behind () is to next show the existence of positive
integers
for which

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).