Definition:Order of Zero

Definition
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 $x$.