# 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* (provided by Runge's theorem) 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 (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

Here are some examples of pole expansions of meromorphic functions:

## See also

## References

- 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 links

- Hazewinkel, Michiel, ed. (2001) [1994], "Mittag-Leffler theorem",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - "Mittag-Leffler's theorem".
*PlanetMath*.