Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / B / 8 / i

The two holomorphic function functions agree throughout their overlap region strip, so they define a single holomorphic function on the union of their domains, which is the whole complex plane. The resulting entire function is bounded outside a sufficiently large disk by the assumptions at infinity. It is bounded on the disk by continuity and compactness. Liouville theorem therefore makes it constant:Agreement on an open overlap region is what permits analytic continuation; equality of unspecified boundary values alone would need additional hypotheses.

First state the normalization needed for the displayed logarithmic factorization: choose a continuous holomorphic function branch $L=\log F$ in the strip that tends to zero at both horizontal ends and has sufficient decay for its Cauchy integrals. In particular the index of the kernel along a horizontal line must be zero.

Choose $\alpha <c_{+}<c_{-}<\beta$ and let ${\Gamma }_{\pm }=\{\xi +i{c}_{\pm }:\xi \in \mathbb{R}\}$, both oriented from left to right. PutThe first is analytic above the lower line and the second below the upper line. On the region between the lines, closing a rectangle gives $C_{+}L-C_{-}L=L$: the lower horizontal side is traversed forward and the upper side backward. HenceThe equation $F{w}_{+}=w_{-}$ becomes $F_{+}{w}_{+}=F_{-}{w}_{-}$ on the overlap region. With the canonical bounded factors normalized at infinity, these two products glue to a bounded entire function. Part (a) givesThis is Wiener-Hopf factorization with the quotient convention used in this question.

The source hypotheses require a qualification: nonvanishing in the strip and $F\to 1$ do not by themselves force index zero or a logarithm tending to zero at both ends. For exampleis analytic and nonzero for $|\mathrm{Im}z|<1/2$, tends to one at both ends, but winds once around zero along the real line. A logarithm continued along that line changes by $2\pi i$, so it cannot have the stated normalization at both ends. One must first separate the index factor, or allow appropriately subtracted factors and a correspondingly different common-function growth class. The displayed constant conclusion follows under the canonical zero-index assumptions above; it is not an unconditional consequence of the two printed properties of $F$. This is the issue measured by the Index of a Wiener-Hopf kernel.

Use the same normalized factors and contours as in part (b), and put $H=G{F}_{-}$. Multiplying the inhomogeneous equation by $F_{-}$ givesThe additive Cauchy decomposition has $C_{+}H-C_{-}H=H$ on the overlap region. Subtract it from this equation:Under the same convergence and boundedness assumptions, these expressions are analytic on complementary half-planes, agree on their overlap region, and glue to a bounded entire function. ThusIf the decay is too weak for an unsubtracted Cauchy integral, the subtraction and resulting common-function growth condition must be specified. Merely asserting pointwise $G\to 0$ is not a substitute for the needed integral transform convergence.