# closed curve theorem

Let $U\subset \u2102$ be a simply connected domain, and suppose $f:U\u27f6\u2102$ is holomorphic. Then

$${\int}_{C}f(z)\mathit{d}z=0$$ |

for any smooth closed curve^{} $C$ in $U$.

More generally, if $U$ is any domain, and ${C}_{1}$ and ${C}_{2}$ are two homotopic^{} smooth closed curves in $U$, then

$${\int}_{{C}_{1}}f(z)\mathit{d}z={\int}_{{C}_{2}}f(z)\mathit{d}z.$$ |

for any holomorphic function $f:U\u27f6\u2102$.

Title | closed curve theorem |
---|---|

Canonical name | ClosedCurveTheorem |

Date of creation | 2013-03-22 12:04:43 |

Last modified on | 2013-03-22 12:04:43 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 7 |

Author | djao (24) |

Entry type | Theorem |

Classification | msc 30E20 |

Related topic | CauchyIntegralTheorem |