# Mittag-Leffler’s theorem

Let $G$ be an open subset of $\u2102$, let $\{{a}_{k}\}$ be a sequence of distinct points in $G$ which has no limit point^{} in $G$. For each $k$, let
${A}_{1k},\mathrm{\dots},{A}_{{m}_{k}k}$ be arbitrary complex coefficients, and define

$${S}_{k}(z)=\sum _{j=1}^{{m}_{k}}\frac{{A}_{jk}}{{(z-{a}_{k})}^{j}}.$$ |

Then there exists a meromorphic function $f$ on $G$ whose poles are exactly the points $\{{a}_{k}\}$ and such that the singular part of $f$ at ${a}_{k}$ is ${S}_{k}(z)$, for each $k$.

Title | Mittag-Leffler’s theorem |
---|---|

Canonical name | MittagLefflersTheorem |

Date of creation | 2013-03-22 13:15:15 |

Last modified on | 2013-03-22 13:15:15 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 4 |

Author | Koro (127) |

Entry type | Theorem |

Classification | msc 30D30 |

Related topic | WeierstrassFactorizationTheorem |