I would like to prove that if the complex-valued function $f(t,z)$, defined for $a \leq t \leq b$ and $ z \in D$ (where $D$ is some domain), is continuous and analytic for each fixed $t$ (for $z \in D$), then $$F(z) = \int_a^b f(t,z)dz, \hspace{2mm} z\in D $$ is analytic on $D$.
So first I let $R$ be a closed rectangle in $D$. By Cauchy's theorem, $$\int_{\partial R} f(t,z) dz = 0, $$ which implies that $$\int_a^b \int_{\partial R} f(t,z)dz \hspace{0.5mm} dt = 0. $$
Now, if I could simply change the order of integration, I would have that $$\int_{\partial R} \int_a^b f(t,z) dt\hspace{0.5mm} dz = \int_{\partial R} F(z) dz = 0. $$ Then by Morera's theorem, since $F(z)$ is continuous, I would be done. However, I don't know if I can change the order, or what determines if I can change the order.
I know there is a theorem called Fubini's theorem that relates to changing orders of integration, but I don't know how to apply it in this case...
When you say $f$ is continuous I suppose you mean that it is jointly continuous in $t$ and $z$. The set $[a,b]\times \partial R$ is a compact set, so $f$ is bounded on this set. This implies integrability on the product and Fubini's Theorem can be applied.