Wiener-Hopf equation ($K(k)F^{-}(k)+F^{+}(k)=R(k)$)

A scalar Wiener-Hopf equation couples two Half-range Fourier transforms analytic in complementary half-planes. After dividing by a kernel factor, additive splitting separates the known forcing. Source poles, boundary terms and contour prescriptions are part of the problem data.

The scalar coefficient multiplying an unknown half-line transform in a Wiener-Hopf equation is its kernel. This is a transform-domain function, distinct from the null space of a linear map. Its analytic zeros, poles and winding number govern Wiener-Hopf factorization. For mixed scalar Helmholtz half-plane diffraction a typical kernel is the outgoing vertical wavenumber $\sqrt{k^2-{\alpha }^2}$.

A forcing pole can be split between analytic half-planes by subtracting a factor's value at the pole. For instance,The second quotient has a removable pole and belongs to the plus expression; the first keeps the prescribed minus-side forcing pole. After Wiener-Hopf factorization, analytic continuation identifies an entire remainder, whose value is fixed by growth and edge conditions. The pole prescription must be stated together with the Fourier transform convention.