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

For a nonvanishing kernel tending to the same nonzero value at both ends, the index counts its winding as the contour is traversed from left to right. Index zero permits a continuous logarithm tending to the same normalization at both ends. Canonical bounded Wiener-Hopf factorization and a constant common entire remainder require this qualification. For example $F(z)=(z+i)/(z-i)$ has index $-1$ on the real line; the bounded half-plane functions $w_{+}=(z-i)/(z+i)$ and $w_{-}=1$ obey $F{w}_{+}=w_{-}$. The analytic factors $F_{+}=z+i$, $F_{-}=z-i$ make their common product $z-i$, demonstrating why bounded input functions alone do not guarantee a constant factored remainder.