Definition:Holomorphic Function/Complex Plane

Definition
Let $U \subseteq \C$ 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 $U$.

We also say that $f$ is complex-differentiable in $U$.

Also defined as
A holomorphic function is sometimes defined as continuously differentiable on the open set $U$.

By Holomorphic Function is Continuously Differentiable, the two definitions are equivalent.

Also known as
Some authors refer to a holomorphic function as a (complex) analytic function.

By Holomorphic Function is Analytic, the two are equivalent.