Wiener–Hopf method

Summary

The Wiener–Hopf method solves Wiener–Hopf equations and boundary value problems whose conditions change type across a half-line. It takes Fourier transforms, so that functions supported on a half-line become functions holomorphic in a half-plane. The key step factors the symbol of the problem into two factors. One is holomorphic and nonzero in the upper half-plane, the other in the lower half-plane. An argument with Liouville's theorem then pins down the solution. It was introduced by Norbert Wiener and Eberhard Hopf for integral equations on a half-line. It is a standard tool in diffraction theory, radiative transfer and fracture mechanics.