TITLE: Holomorphic Parameter Integral
QUESTION [8 upvotes]: Let $U\subseteq\mathbb{C}$ be open, $\gamma$ a way in $\mathbb{C}$ which is picewise differentiable continiously and $f\colon rg(\gamma)\times U\to\mathbb{C}$ a continious function. Consider the parameter integral
    $$
F(z):=\int_{\gamma}f(\omega,z), d\omega, z\in U.
$$
    Show: Let the function $z\mapsto f(\omega,z)$ be holomorphic in $U$ for every $\omega\in rg(\gamma)$ with continious derivation $\frac{\partial}{\partial z}f(\omega,z)$ on $rg(\gamma)\times U$. Then $F(z)$ is holomorphic in $U$, and one can differentiate under the integral:
    $$
F'(z)=\int_{\gamma}\frac{\partial}{\partial z}f(\omega,z)\, d\omega, z\in U
$$


Unfortunately I do not know how to show that.
I know the proof when $f$ is a function with values in $\mathbb{R}$ but cannot proof it here when $f$ has values in $\mathbb{C}$.

REPLY [2 votes]: Using Cauchy's integral formula , we have$$f(w,z)=\frac{1}{2\pi i} \oint_C \frac{f(w,\zeta)}{\zeta-z}\, \mathrm{d} \zeta,$$
where $C$ is the boundary of a disc contained in $U$, which encloses $z$.
Therefore
$$F(z)=\frac{1}{2\pi i}\int_\gamma \oint_C \frac{f(w,\zeta)}{\zeta-z}\, \mathrm{d} \zeta \mathrm{d} w. $$
Using Fubini's theorem we can change the order of integration, so that
$$F(z)=\oint_C \left[ \frac{1}{2\pi i} \int_\gamma f(w,\zeta)\, \mathrm{d}w \right] \frac{\mathrm{d} \zeta}{\zeta-z}. $$
Integrals of the form $\oint_C \frac{\varphi(\zeta)}{\zeta-z} \mathrm{d} \zeta$, with continuous $\varphi$ can be differentiated under the integral symbol (cf. Lemma 3 in Chapter 4 of Ahlfors' text).
Thus
$$F'(z)=\oint_C \left[ \frac{1}{2\pi i} \int_\gamma f(w,\zeta)\, \mathrm{d}w \right] \frac{\mathrm{d} \zeta}{(\zeta-z)^2}, $$
and applying Fubini's theorem once more we find
$$F'(z)= \int_\gamma \left[ \frac{1}{2 \pi i} \oint_C \frac{f(w,z)}{(\zeta-z)^2} \mathrm{d} \zeta \right] \mathrm{d} w $$
We recognize the inner integral as $\frac{\partial f}{\partial z}$ (Cauchy's integral formula). Overall we have shown
$$F'(z)=\int_\gamma \frac{\partial}{\partial z} f(w,z) \mathrm{d} w $$