# Laurent series

A *Laurent series ^{}* centered about $a$ is a series of the form

$$\sum _{k=-\mathrm{\infty}}^{\mathrm{\infty}}{c}_{k}{(z-a)}^{k}$$ |

where ${c}_{k},a,z\in \u2102$. The *principal part* of a Laurent series is the subseries ${\sum}_{k=-\mathrm{\infty}}^{-1}{c}_{k}{(z-a)}^{k}$.

One can prove that the above series converges everywhere inside the (possibly empty) set

$$ |

where

$${R}_{1}:=\underset{k\to \mathrm{\infty}}{lim\; sup}{|{c}_{-k}|}^{1/k}$$ |

and

$${R}_{2}:=1/\left(\underset{k\to \mathrm{\infty}}{lim\; sup}{|{c}_{k}|}^{1/k}\right).$$ |

Every Laurent series has an associated function, given by

$$f(z):=\sum _{k=-\mathrm{\infty}}^{\mathrm{\infty}}{c}_{k}{(z-a)}^{k},$$ |

whose domain is the set of points in $\u2102$ on which the series converges. This function is analytic inside the annulus $D$, and conversely, every analytic function on an annulus is equal to some unique Laurent series. The coefficients of Laurent series for an analytic function can be determined using the Cauchy integral formula^{}.

Title | Laurent series |
---|---|

Canonical name | LaurentSeries |

Date of creation | 2013-03-22 12:04:52 |

Last modified on | 2013-03-22 12:04:52 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 12 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 30B10 |

Synonym | Laurent expansion |

Related topic | EssentialSingularity |

Related topic | CoefficientsOfLaurentSeries |

Defines | principal part |