Mittag-Leffler's Partial Fractions Theorem

Let any finite or infinite set of points having no finite limit point be prescribed and associate with each of its points a principal part, i.e., a rational function of the special form

for , 2, ..., . Then there exists a meromorphic function which has poles with the prescribed principal parts at precisely the prescribed points, and is otherwise regular. It can be represented in the form of a partial fraction decomposition from which one can read off again the poles, along with their principal parts. Further, if  is one such function, then

is the most general function satisfying the conditions of the problem, where  denotes an arbitrary entire function.

REFERENCES:

Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, pp. 37-39, 1996.

Krantz, S. G. "The Mittag-Leffler Theorem." §8.3.6 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 112-113, 1999.