Holomorphic complex parameter integrals

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Maybe it is a basic question, but I can not find an answer in standard textbooks or at StackExchange.

Let $D\subseteq \mathbb C$ be an open set and $\Omega\subseteq \mathbb C^n$. Further, let $f:D\times \Omega \rightarrow \mathbb C$ be a function, which is holomorphic with respect to its first variable $\eta\in D$ and $f(\eta,x)$ absolutely integrable on $\Omega$ with respect to its second variable for all $\eta\in D$. What are the conditions that also the parameter integral $$ F(\eta) = \int_\Omega dx f(\eta,x) $$ is holomorphic in $\eta\in D$? What are the conditions if one replace "holomorphic" by "meromorphic"?

I would guess, that in order to use Cauchy-Riemann equations, one also has to claim the deriviatives $\frac{\partial f_1}{\partial\eta_1}$ and $\frac{\partial f_1}{\partial\eta_2}$ to be absolutely integrable?