Mittag-Lefler theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $z_n$ is a sequence of points in $\mathbb{C}$ that form a discrete closed subset of $\mathbb{C}$ (in particular, if there are infinitely many, $\left|z_n\right|\to \mathrm{\infty }$). Suppose $P_n$ is a sequence of nonzero polynomials. Then, there exists a meromorphic function $f$ on the whole of $\mathbb{C}$ such that the poles of $f$ are precisely at the $z_n$s, and moreover, such that for each $z_n$, $P(1/(z-z_n))$ is the principal part of $f$ at $z_n$.

Importance

The Mittag-Lefler theorem states that Cousin's additive problem can be solved for sections over discrete closed subsets of $\mathbb{C}$.