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.

TheoremEdit

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 (provided by Runge's theorem) without changing the principal parts of the   and in such a way that convergence is guaranteed.

ExampleEdit

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 functionsEdit

Here are some examples of pole expansions of meromorphic functions:

 
 
 
 
 
 
 

See alsoEdit

ReferencesEdit

  • Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1.
  • Conway, John B. (1978), Functions of One Complex Variable I (2nd ed.), Springer-Verlag, ISBN 0-387-90328-3.

External linksEdit