Let $\gamma$ be a curve in $\mathbb{C}$, and let $\gamma_0$ be a circle in an open connected set $A \subset \mathbb{C}$ around $z_0 \in A$. Suppose the interior of $\gamma_0$ lies in $A$. Let $z$ be inside $\gamma_0$. Assume $n(\gamma_0; z) = 1$, and $f(z, w)$ is a continuous function of $z$, $w$ for $z$ in $A$ and $w$ on $\gamma$. For each $w$ on $\gamma$ assume that $f$ is analytic in $z$.
Consider $$F(z) = \frac{1}{2\pi i} \int_\gamma \int_{\gamma_0} \frac{f(\zeta, w)}{\zeta - z} d\zeta dw.$$ Marsden (Basic Complex Analysis, p. 179) claims that in terms of real integrals $F(z)$ has the form $$\int_a^b \int_\alpha^\beta h(x, y) dx dy + i\int_a^b \int_\alpha^\beta k(x, y) dx dy.$$
Can someone explain step-by-step why it is so?