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,

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

Title	Gauss’ mean value theorem
Canonical name	GaussMeanValueTheorem
Date of creation	2013-03-22 13:35:33
Last modified on	2013-03-22 13:35:33
Owner	Johan (1032)
Last modified by	Johan (1032)
Numerical id	12
Author	Johan (1032)
Entry type	Theorem
Classification	msc 30E20
Related topic	GaussMeanValueTheoremForHarmonicFunctions
Related topic	AverageValueOfFunction