I have the following question in my complex analysis class:
Let $\gamma: [0,1] \rightarrow \mathbb{C}$ be a path and $G \subseteq \mathbb{C}$ a domain. Furthermore let $f: \operatorname{tr}(\gamma) \cdot G\rightarrow \mathbb{C}$ continuous, so that the partial function $z\mapsto f(w,z)\quad \forall w \in \operatorname{tr}(\gamma)$ is holomorphic. Consider $F: G\rightarrow \mathbb{C}, \quad F(z) = \int_\gamma f(w,z) \, dw.$
Show that:
1.) $F$ holomorphic on $G$ by using Morera's and Fubini's Theorem.
2.) $\displaystyle F^{(k)}(z)=\int_\gamma \frac{{\partial}^k f}{\partial z^k}(w,z) \, dw$ for all $z \in G$ and $k \in \mathbb{N}.$
My problem is that we never really used the theorems above, but I have the feeling that it will just calculating using the theorems. Furthermore, for the second question help would be highly appreciated.