Definition:Holomorphic Function

Complex Function
Let $U\subset\C$ be an open set.

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

Then $f$ is holomorphic in $U$ $f$ is differentiable at each point of $U$.

Also known as
A holomorphic function on an open set of $\C$ is known as a (complex) analytic function.

Also see

 * Definition:Complex Analytic Function