Gauss’ mean value theorem for harmonic functions

If the functionu(z)u(x,y)  is harmonic in a domain of complex plane which contains the disc  |z-z0|r,  then

u(z0)=12π02πu(z0+reiφ)𝑑φ. (1)

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

Title Gauss’ mean value theorem for harmonic functionsPlanetmathPlanetmath
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