The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex-analytical properties of transformed functions. Typically, the standard Fourier transform is used, but examples exist using other transforms, such as the Mellin transform. In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the complex plane, typically, a thin strip containing the real line. Analytic continuation guarantees that these two functions define a single function analytic in the entire complex plane, and Liouville's theorem implies that this function is an unknown polynomial, which is often zero or constant. Analysis of the conditions at the edges and corners of the boundary allows one to determine the degree of this polynomial.
Wiener–Hopf decomposition The fundamental equation that appears in the Wiener-Hopf method is of the form
A ( α ) Ξ + ( α ) + B ( α ) Ψ − ( α ) + C ( α ) = 0 , {\displaystyle A(\alpha )\Xi _{+}(\alpha )+B(\alpha )\Psi _{-}(\alpha )+C(\alpha )=0,}
where A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} are known holomorphic functions, the functions Ξ + ( α ) {\displaystyle \Xi _{+}(\alpha )} , Ψ − ( α ) {\displaystyle \Psi _{-}(\alpha )} are unknown and the equation holds in a strip τ − < I m ( α ) < τ + {\displaystyle \tau _{-}<{\mathfrak {Im}}(\alpha )<\tau _{+}} in the complex α {\displaystyle \alpha } plane. Finding Ξ + ( α ) {\displaystyle \Xi _{+}(\alpha )} , Ψ − ( α ) {\displaystyle \Psi _{-}(\alpha )} is what's called the Wiener-Hopf problem. The key step in many Wiener–Hopf problems is to decompose an arbitrary function Φ {\displaystyle \Phi } into two functions Φ ± {\displaystyle \Phi _{\pm }} with the desired properties outlined above. In general, this can be done by writing
Φ + ( α ) = 1 2 π i ∫ C 1 Φ ( z ) d z z − α {\displaystyle \Phi _{+}(\alpha )={\frac {1}{2\pi i}}\int _{C_{1}}\Phi (z){\frac {dz}{z-\alpha }}}
and
Φ − ( α ) = − 1 2 π i ∫ C 2 Φ ( z ) d z z − α , {\displaystyle \Phi _{-}(\alpha )=-{\frac {1}{2\pi i}}\int _{C_{2}}\Phi (z){\frac {dz}{z-\alpha }},}
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