Argument principle

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

This fact is an application of the following pivotal fact/result/idea: residue theorem
View other applications of residue theorem OR Read a survey article on applying residue theorem

Statement

Suppose $U\subset \mathbb {C}$ is an open subset and $c$ is a 0-homologous cycle in $U$. Suppose $f$ is a meromorphic function on $U$ such that no zero or pole of $f$ lies in $U$. Then we have:

$n(f\circ c;0)=\sum ord(z_{j})n(c;z_{j})$

Where the sum is taken over all zeros/poles $z_{j}$ for $f$ and $ord(z_{j})$ is the unique integer $n$ such that $(z-z_{j})^{n}f(z)$ has neither a zero nor a pole at $z_{j}$.

Equivalently, we can write this as:

$n(f\circ c;0)=\sum n(c;z_{j})-\sum n(c;z_{j})$

where the first summation is for zeros, counted with multiplicity, and the second summation is for poles, counted with multiplicity.