\begin{definition}[Definition:Order of Zero]
Let $f: \C \to \C$ be a complex function.
Let $U \subset \C$ be such that $f$ is analytic in $U$.
Let $x \in U$ be a zero of $f$.
That is, let $x$ be such that $\map f x = 0$.
Let $n \in \Z_{\ge 0}$ be the least positive integer such that:
:$\map {f^{\paren n} } x \ne 0$
where $f^{\paren n}$ denotes the $n$th derivative of $f$.
Then $n$ is the '''order of the zero''' at $x$.
\end{definition}