# Gauss’ mean value theorem

Let $\Omega$ be a domain in $\mathbb{C}$ and suppose $f$ is an analytic function on $\Omega$. Furthermore, let $C$ be a circle inside $\Omega$ with center $z_{0}$ and radius $r$. Then $f(z_{0})$ is the mean value of $f$ along $C$, that is,

 $f(z_{0})=\frac{1}{2\pi}\int_{0}^{2\pi}f(z_{0}+re^{i\theta})d\theta.$
Title Gauss’ mean value theorem GaussMeanValueTheorem 2013-03-22 13:35:33 2013-03-22 13:35:33 Johan (1032) Johan (1032) 12 Johan (1032) Theorem msc 30E20 GaussMeanValueTheoremForHarmonicFunctions AverageValueOfFunction