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Mittag-Leffler's theorem (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_kk}$ be arbitrary complex coefficients, and define

$\displaystyle S_k(z) = \sum_{j=1}^{m_k} \frac{A_{jk}}{(z-a_k)^j}.$
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$.



"Mittag-Leffler's theorem" is owned by Koro.
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See Also: Weierstrass factorization theorem


Attachments:
table of partial fraction expansions (Example) by rspuzio
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Cross-references: singular, poles, function, meromorphic, coefficients, complex, limit point, points, sequence, open subset
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This is version 1 of Mittag-Leffler's theorem, born on 2002-12-11.
Object id is 3732, canonical name is MittagLefflersTheorem.
Accessed 4141 times total.

Classification:
AMS MSC30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

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