# Definition:Analytic Function/Complex Plane

< Definition:Analytic Function(Redirected from Definition:Analytic Complex Function)

## Contents

## Definition

Let $U \subset \C$ be an open set.

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

Then $f$ is **analytic** in $U$ if and only if for every $z_0 \in U$ there exists a sequence $(a_n) : \N \to \C$ such that the series:

- $\displaystyle \sum_{n\mathop = 0}^\infty a_n(z-z_0)^n$

converges to $f(z)$ in a neighborhood of $z_0$ in $U$.

## Also known as

An **analytic complex function** is also referred to as a **holomorphic function**.

## Also see

- Definition:Holomorphic Function
- Definition:Meromorphic Function: a complex function which is analytic except at a countable number of isolated points.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**analytic**(of a complex function)