Definition:Differentiable Mapping/Complex Function/Point
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Definition
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 $z_0$.
Sources
- 1973: John B. Conway: Functions of One Complex Variable: Chapter $III$: Elementary Properties and Examples of Analytic Functions: $\S2$: Analytic Functions: Definition $2.1$
- 2001: Christian Berg: Kompleks funktionsteori: $\S 1.1$