Gauss’ mean value theorem

Let $\mathrm{\Omega }$ be a domain in $ℂ$and suppose $f$ is an analytic function on $\mathrm{\Omega }$.Furthermore, let $C$ be a circle inside $\mathrm{\Omega }$with center $z_0$ and radius $r$. Then $f(z_0)$is the mean value of $f$ along $C$, that is,

$$f(z_0)=\frac{1}{2\pi }{\int }_0^{2\pi }f(z_0+r{e}^{i\theta })𝑑\theta .$$