# Cauchy residue theorem

Let $U\subset \u2102$ be a simply connected domain, and suppose $f$ is a complex valued function which is defined and analytic on all but finitely many points ${a}_{1},\mathrm{\dots},{a}_{m}$ of $U$. Let $C$ be a closed curve^{} in $U$ which does not intersect any of the ${a}_{i}$. Then

$${\int}_{C}f(z)\mathit{d}z=2\pi i\sum _{i=1}^{m}\eta (C,{a}_{i})\mathrm{Res}(f;{a}_{i}),$$ |

where

$$\eta (C,{a}_{i}):=\frac{1}{2\pi i}{\int}_{C}\frac{dz}{z-{a}_{i}}$$ |

is the winding number of $C$ about ${a}_{i}$, and $\mathrm{Res}(f;{a}_{i})$ denotes the residue of $f$ at ${a}_{i}$.

The Cauchy residue theorem generalizes both the Cauchy integral theorem (because analytic functions have no poles) and the Cauchy integral formula^{} (because $f(x)/{(x-a)}^{n}$ for analytic $f$ has exactly one pole at $x=a$ with residue $\mathrm{Res}(f(x)/{(x-a)}^{n},a)={f}^{(n)}(a)/n!)$.

Title | Cauchy residue theorem |
---|---|

Canonical name | CauchyResidueTheorem |

Date of creation | 2013-03-22 12:04:58 |

Last modified on | 2013-03-22 12:04:58 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 10 |

Author | djao (24) |

Entry type | Theorem |

Classification | msc 30E20 |

Synonym | Cauchy residue formula |

Synonym | residue theorem |

Related topic | Residue |

Related topic | CauchyIntegralFormula |

Related topic | CauchyIntegralTheorem |