# 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

 $\displaystyle u(z_{0})=\frac{1}{2\pi}\int_{0}^{2\pi}u(z_{0}+re^{i\varphi})\,d\varphi.$ (1)

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

Title Gauss’ mean value theorem for harmonic functions GaussMeanValueTheoremForHarmonicFunctions 2013-03-22 14:57:39 2013-03-22 14:57:39 PrimeFan (13766) PrimeFan (13766) 8 PrimeFan (13766) Theorem msc 31A05 msc 30F15 GaussMeanValueTheorem