In algebraic geometry, for a coherent sheaf $\mathcal{F}$ on a variety $X$ and an irreducible component $Z$ of $\mathrm{supp}(\mathcal{F})$, the multiplicity of $\mathcal{F}$ in $Z$ is $$ \mathrm{Length}_{\mathcal{O}_{X,\xi}}\mathcal{F}_\xi, $$ where $\mathcal{O}_X$ is the structure sheaf of $X$ and $\xi$ is the generic point of $Z$.
My question is: what is the analogous notion in the context of complex analytic geometry?
More precisely, for a coherent analytic sheaf $\mathcal{F}$ on a complex analytic space $X$ and an analytic irreducible component $Z$ of $\mathrm{supp}(\mathcal{F})$, what is the multiplicity of $\mathcal{F}$ in $Z$?