# Definition:Multiplicity (Complex Analysis)

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## Definition

Let $f : \C \to \C$ be a function.

Suppose there is $a \in \C$ such that $f(a) = 0$.

Then $a$ is said to be a **zero of multiplicity $k$** if there exists non-zero $L \in \R$ such that:

- $\displaystyle \lim_{z \to a} \frac{|f(z)|}{|z-a|^k} = L$