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?


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 $K$.


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