= Multiplicity of a zero
{title2=$\operatorname{ord}_{z_0}f$}
= Zero multiplicity
{synonym}
A <holomorphic function> that is not identically zero has a zero of multiplicity $m$ at $z_0$ if $f(z)=(z-z_0)^m h(z)$ for a <holomorphic function> $h$ with $h(z_0)\ne0$. Its <Taylor series> identifies $m$ as the first nonzero coefficient index. Logarithmic differentiation gives $f'/f=m/(z-z_0)+h'/h$, so the <residue> of $f'/f$ is $m$. Counting zeros with <zero multiplicity> repeats that zero $m$ times.
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