Mittag-Leffler's theorem

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In complex analysis, Mittag-Leffler's theorem concerns the existence of meromorphic functions with prescribed poles. Conversely, it can be used to express any meromorphic function as a sum of partial fractions. 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 has only a removable singularity 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]