# Definition:Multiplicity (Complex Analysis)

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