Order of a zero or pole

Summary

The order of a meromorphic function at a point $a$ is the integer $n$ such that it behaves like $c (z - a)^n$ near $a$, with $c \neq 0$. See the definition. At a zero it is the order of the zero. At a pole it is minus the order of the pole. This single integer turns products of meromorphic functions into sums, like a logarithm.