Definition:Order of Zero

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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 $x$.

Simple Zero

Let the order of the zero $x$ be $1$.

Then $x$ is a simple zero.