Definition:Analytic Function/Complex Plane

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

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

Then $f$ is analytic in $U$ 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 as the word regular is overused.

Also see

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