Wiener-Hopf kernel ($K(\alpha )$)

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 $\sqrt{k^2-{\alpha }^2}$.