The Mittag-Leffler Theorem states that given any discrete set of points in the complex plane together with a prescribed singular part at each point, there exists a meromorphic function on the whole plane having poles at exactly those points with exactly those singular parts. Named for Gosta Mittag-Leffler, it is a foundational existence result of complex analysis, the counterpart for poles to the Weierstrass factorization theorem's treatment of zeros.
Facts
StatementFor an open set U in the complex plane and a subset E of U whose limit points, if any, occur only on the boundary of U, there exists a meromorphic function on U with poles exactly at the points of E and prescribed principal parts there. 2 Classification
Statement Form Sources
1. Wikipedia: Mittag-Leffler's theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles.
2. Mittag-Leffler Theorem (Wikipedia)
Wikimedia FoundationTheorem statement section, opening sentence
Let U be an open set in ℂ and E ⊂ U be a subset whose limit points, if any, occur on the boundary of U.
Lead section, naming sentence
The theorem is named after the Swedish mathematician Gosta Mittag-Leffler who published versions of the theorem in 1876 and 1884.
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