# Mittag-Leffler’s theorem

Let $G$ be an open subset of $\mathbb{C}$, let $\{a_{k}\}$ be a sequence of distinct points in $G$ which has no limit point in $G$. For each $k$, let $A_{1k},\dots,A_{m_{k}k}$ 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 MittagLefflersTheorem 2013-03-22 13:15:15 2013-03-22 13:15:15 Koro (127) Koro (127) 4 Koro (127) Theorem msc 30D30 WeierstrassFactorizationTheorem