 Solution

ID: past-exam-of-the-perturbation-methods-and-applied-complex-variables-course-of-the-university-of-oxford/2008/b/8/i/c/solution

Use the same normalized factors and contours as in part (b), and put

H = G F_{-}

. Multiplying the inhomogeneous equation by

F_{-}

gives

F_{+} w_{+} - F_{-} w_{-} = H .

The additive Cauchy decomposition has

C_{+} H - C_{-} H = H

on the overlap region. Subtract it from this equation:

F_{+} w_{+} - C_{+} H = F_{-} w_{-} - C_{-} H .

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. Thus

F_{+} w_{+} - \frac{1}{2 πi} \int_{Γ_{+}} \frac{G ( ζ ) F _{-} ( ζ )}{ζ - z} d ζ = F_{-} w_{-} - \frac{1}{2 πi} \int_{Γ_{-}} \frac{G ( ζ ) F _{-} ( ζ )}{ζ - z} d ζ = C .

If 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.

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