# identity theorem of holomorphic functions

If the functions $f$ and $g$ are holomorphic in a domain $D$ of the complex plane and the equation

$f(z)=g(z)$ | (1) |

is true in an infinite subset $S$ of $D$ having an accumulation point^{} ${z}_{0}$ in $D$, then (1) is true in the whole $D$.

Remark. The subset $S$ may be e.g. some neighbourhood of ${z}_{0}$ or some arc containing ${z}_{0}$.

Title | identity theorem of holomorphic functions |
---|---|

Canonical name | IdentityTheoremOfHolomorphicFunctions |

Date of creation | 2013-03-22 16:47:05 |

Last modified on | 2013-03-22 16:47:05 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 11 |

Author | rspuzio (6075) |

Entry type | Theorem |

Classification | msc 30A99 |

Synonym | rigidity theorem for analytic functions |

Related topic | IdentityTheoremOfPowerSeries |

Related topic | IdentityTheorem |