# Definition:Multiplicity (Complex Analysis)

(Redirected from Definition:Multiplicity (Analytic Function))

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It has been suggested that this page or section be merged into Definition:Order of Zero.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## 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 and only if there exists non-zero $L \in \R$ such that:

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

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