Gauss’ mean value theorem for harmonic functions

If the function  $u(z)\equiv u(x,y)$  is harmonic in a domain of complex plane which contains the disc  $|z-z_0|\leqq r$,  then

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

Conversely, if a real function $u(x,y)$ is continuous in a domain $G$ of ${ℝ}^2$ and satisfies on all circles of $G$ the equation (1), then it is harmonic.

Title	Gauss’ mean value theorem for harmonic functions
Canonical name	GaussMeanValueTheoremForHarmonicFunctions
Date of creation	2013-03-22 14:57:39
Last modified on	2013-03-22 14:57:39
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	8
Author	PrimeFan (13766)
Entry type	Theorem
Classification	msc 31A05
Classification	msc 30F15
Related topic	GaussMeanValueTheorem