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, $|z_{n}|\to \infty$). 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 $\mathbb {C}$.