Definition:Order of Zero

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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 $f \left({x}\right) = 0$.


Let $n \in \Z_{\ge 0}$ be the least positive integer such that:

$f^{(n)} \left({x}\right) \ne 0$

where $f^{(n)}$ 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.