Definition:Analytic Function/Complex Plane

Definition
Let $f \left({z}\right): \C \to \C$ be a single-valued continuous complex function in a domain $D \subseteq \C$.

Let $f$ be complex-differentiable in $D$.

Then $f \left({z}\right)$ is described as being analytic on $D$.

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

Also see

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