# 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

$u({z}_{0})={\displaystyle \frac{1}{2\pi}}{\displaystyle {\int}_{0}^{2\pi}}u({z}_{0}+r{e}^{i\phi})\mathit{d}\phi .$ | (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^{} |
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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 |