Definition:Differentiable Mapping/Complex Function/Point

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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.

This limit, if it exists, is called the derivative of $f$ at $z_0$.