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 $(a_n) : \N \to \C$ 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.