Mittag-Leffler's theoremMittagLefflersTheorem2002-12-11 09:50:392002-12-11 09:50:39Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $G$ 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
$A_{1k},\dots,A_{m_kk}$ 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 $k$.