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 $\map f a = 0$.

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

$\displaystyle \lim_{z \mathop \to a} \dfrac {\cmod {\map f z} } {\cmod {z - a}^k} = L$