Definition:Differentiable Mapping/Complex Function

Definition
Let $f \left({z}\right): \C \to \C$ be a single-valued continuous complex function in a domain $D \subseteq \C$.

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

Then $f \left({z}\right)$ is complex-differentiable at $z_0$ iff the limit:
 * $\displaystyle \lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} h$

exists as a finite number and is independent of how the complex increment $h$ tends to $0$.

If such a limit exists, it is called the derivative of $f$ at $z_0$.

If $f \left({z}\right)$ is complex-differentiable at every point in $D$, it is differentiable in $D$. Such a function is called analytic.