Weierstrass's 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 (possibly repeating) complex numbers that does not cluster in $C$ : in other words, $| z_{n} | \to \infty$ if the sequence is infinite. Then, there exists an entire function <math<f:\mathbb{C} \to \mathbb{C}</math> such that for any $z \in C$ , the order of zero for $f$ at $z$ equals the number of times $z$ occurs in the sequence $(z_{n})$ .

Importance

This solves Cousin's multiplicative problem in a particular case over $C$ .