Definition:Analytic Function/Complex Plane

It has been suggested that this page or section be merged into Definition:Holomorphic Function. 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 $U \subset \C$ 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)