Mittag-Leffler's theorem


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

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. 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 without changing the principal parts of the and in such a way that convergence is guaranteed.

Example

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 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 to a meromorphic function with the desired properties.

Pole expansions of meromorphic functions

Here are some examples of pole expansions of meromorphic functions: