Mittag-Leffler's theorem

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In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. It is sister to the Weierstrass factorization theorem, which asserts existence of holomorphic functions with prescribed zeros. It is named after Gösta Mittag-Leffler.

Theorem[edit]

Let be an open set in and a closed discrete subset. For each in , let be a polynomial in . There is a meromorphic function on such that for each , the function is holomorphic at . In particular, the principal part of at is .

One possible proof outline is as follows. Notice that if is finite, it suffices to take . If is not finite, consider the finite sum where is a finite subset of . While the may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of D (provided by Runge's theorem) without changing the principal parts of the and in such a way that convergence is guaranteed.

Example[edit]

Suppose that we desire a meromorphic function with simple poles of residue 1 at all positive integers. With notation as above, letting

and , Mittag-Leffler's theorem asserts (non-constructively) the existence of a meromorphic function with principal part at for each positive integer . This has the desired properties. More constructively we can let

.

This series converges normally on (as can be shown using the M-test) to a meromorphic function with the desired properties.

Pole expansions of meromorphic functions[edit]

Here are some examples of pole expansions of meromorphic functions:

See also[edit]

References[edit]

External links[edit]