m.g. mittag-leffler

Mittag-Leffler's theorem

In complex analysis, Mittag–Leffler's theorem concerns the existence of functions with prescribed zeros or poles. It is named after Gösta Mittag-Leffler.


Let Omega be an open set in mathbb C and EsubsetOmega a discrete subset. For ainmathbb C, write mathcal H(mathbb C-{a}) for the set of all holomorphic functions on mathbb C-{a}. Suppose we are given a function p_ainmathcal H(mathbb C-{a}), for every ain E. Then there exists finmathcal H(Omega-E) such that for all ain E, the function f-p_a is holomorphic at a. In particular, there exists finmathcal H(Omega-E) whose principal parts at the points of E are prescribed.

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