Definition:Differentiable Mapping/Complex Function

Definition

At a Point

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

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

Let $z_0 \in U$ be a point in $U$.

Then $f$ is complex-differentiable at $z_0$ if and only if the limit:

$\ds \lim_{h \mathop \to 0} \frac {\map f {z_0+h} - \map f {z_0}} h$

exists as a finite number.

In an Open Set

Let $U \subseteq \C$ be an open set.

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

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