Definition:Derivative/Complex Function/Point

Definition
Let $D\subseteq \C$ be an open set.

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

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

Let $f$ be complex-differentiable at the point $z_0$.

That is, suppose the limit $\ds \lim_{h \mathop \to 0} \ \frac {\map f {z_0 + h} - \map f {z_0} } h$ exists.

Then this limit is called the derivative of $f$ at the point $z_0$. It can be denoted $f' \left({z_0}\right)$,