# Definition:Analytic Function/Complex Plane

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## 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 $\sequence {a_n}: \N \to \C$ such that the series:

- $\ds \sum_{n \mathop = 0}^\infty a_n \paren {z - z_0}^n$

converges to $\map f z$ in a neighborhood of $z_0$ in $U$.

## Also known as

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

Some sources use the term **regular function**, but this is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ as the word **regular** is overused.

The order of words can be varied: **complex analytic function**.

## Also see

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

- Results about
**analytic complex functions**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**analytic**:**1.**(of a complex function) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**analytic function (holomorphic function, regular function)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**analytic function (holomorphic function, regular function)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**analytic**(of a complex function)