Let $\displaystyle \mathcal U\subset\mathbb C$ be open and consider a function $\displaystyle f:[0,1]\times\mathcal U\longrightarrow\mathbb C$ so that for all $\displaystyle t\in[0,1]$ the function $\displaystyle f(t,\cdot):\mathcal U\longrightarrow\mathbb C$ is analytic. Prove that $\displaystyle F(z)=\int_0^1 f(t,z)\,dt$ is analytic.

Any ideas? Can I use Morera's Theorem? How?

Thanks.