Wiener-Hopf factorization ($K=K^{+}{K}^{-}$)

A scalar Wiener-Hopf factorization writes a transform kernel on its contour as $K=K^{+}{K}^{-}$. The plus factor is holomorphic and nonzero in the designated upper domain, and the minus factor has the corresponding property in the lower domain. Normalizations and any index or polynomial factors must be specified along with these domains.

For real $a,b>0$, a concrete real-line example isThe possible zero and pole of $K^{+}$ lie below the real axis, so $K^{+}$ is holomorphic and nonzero in the complex upper half-plane; those of $K^{-}$ lie above, giving the lower-domain property. Both factors tend to one at infinity, and multiplying them verifies the kernel exactly.

The allocation of zeros, poles and branch cuts is essential in a Wiener-Hopf equation. An outgoing continuation fixes how physical modal poles and square-root branches are assigned; a factor with a zero in its designated analytic domain is not the required nonvanishing factor. Products and quotient conventions should be declared before using a table of factors.