= Solution
{c}
Use the same normalized factors and <contours> as in part (b), and put $H=GF_-$. 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
$$
\boxed{
F_+w_+-\frac1{2\pi i}\int_{\Gamma_+}
\frac{G(\zeta)F_-(\zeta)}{\zeta-z}\,d\zeta
=F_-w_--\frac1{2\pi i}\int_{\Gamma_-}
\frac{G(\zeta)F_-(\zeta)}{\zeta-z}\,d\zeta
=C.}
$$
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 $G\to0$ is not a substitute for the needed <integral transform> convergence.
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