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.

The theorem is named after the Swedish mathematician Gösta Mittag-Leffler who published versions of the theorem in 1876 and 1884.

be an open set in

be a subset whose limit points, if any, occur on the boundary of

without constant coefficient, i.e. of the form

Then there exists a meromorphic function

whose poles are precisely the elements of

has only a removable singularity at

; in particular, the principal part of

Furthermore, any other meromorphic function

with these properties can be obtained as

is an arbitrary holomorphic function on

One possible proof outline is as follows.

is finite, it suffices to take

is a finite subset of

may not converge as F approaches E, one may subtract well-chosen rational functions with poles outside of

(provided by Runge's theorem) without changing the principal parts of the

and in such a way that convergence is guaranteed.

Suppose that we desire a meromorphic function with simple poles of residue 1 at all positive integers.

With notation as above, letting

, Mittag-Leffler's theorem asserts the existence of a meromorphic function

with principal part

for each positive integer

More constructively we can let

This series converges normally on any compact subset of

(as can be shown using the M-test) to a meromorphic function with the desired properties.

Here are some examples of pole expansions of meromorphic functions:

sec ⁡ ( z ) ≡ − csc ⁡