# Gauss’ mean value theorem

Let $\mathrm{\Omega}$ be a domain in $\u2102$
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,

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

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 |