Multiplicity of a zero (${\mathrm{ord}}_{z_0}f$)

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 $h(z_0)\ne 0$. Its Taylor series identifies $m$ as the first nonzero coefficient index. Logarithmic differentiation gives $f^{\prime }/f=m/(z-z_0)+h^{\prime }/h$, so the residue of $f^{\prime }/f$ is $m$. Counting zeros with zero multiplicity repeats that zero $m$ times.