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# 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 $k$.

Related:

WeierstrassFactorizationTheorem

Type of Math Object:

Theorem

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Reference

## Mathematics Subject Classification

30D30*no label found*

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