Mittag-Leffler's theorem

From Wikipedia, the free encyclopedia

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

[edit] Theorem

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