TITLE: Uncountable sum of holomorphic functions is holomorphic?
QUESTION [5 upvotes]: When I have a function $f\left( \alpha ,z \right)$ depending on a parameter $\alpha \in [0,1]$ and $z\in K \subset \mathbb{C}$, where $K$ is a domain on which $f\left( \alpha ,z \right)$ is holomorphic for all $\alpha \in [0,1]$ it's true in general that 
$\int_0^1 f\left( \alpha ,z \right) d \alpha$ is holomorphic again or there are some pathological cases where it's not?

REPLY [2 votes]: It is always holomorphic. This can be shown with Morera's theorem. Let $\Delta$ be a triangle inside $K$. Then (let's denote $F=\int_0^1 f(\alpha, z)d\alpha$)
$$\int_{\partial\Delta} F(\alpha, z) dz = \int_{\partial\Delta}\int_0^1 f(\alpha, z)d\alpha dz$$
$$= \int_0^1 \int_{\partial\Delta} f(\alpha, z) dz d\alpha = \int_0^1 0 d\alpha = 0$$
The order of the integrals can be changed by Fubini's theorem and the integral of $f$ over the triangle's border is $0$ by Cauchy's integral theorem.
Now by Morera's theorem the function $F$ is holomorphic on $K$.