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

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.