Definitions

# 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.

## Theorem

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