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
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