# Complex Analysis/Complex Functions/Complex Derivatives

With the concept of the limit of a complex function now established, we can now introduce the differentiability of a complex function.

2.3.1.: Derivatives

Definition: Let f be a complex valued function defined in a neighborhood of $z_0$. Then the derivative of f at $z_0$ is given by:

$\frac{df}{dz}(z_0)\equiv f'(z_0) := \lim_{\Delta z\rightarrow 0} \frac {f(z_0+ \Delta z) - f(z_0)}{\Delta z}$, where $\Delta z$ is a complex number.

Provided that this limit exists, f is said to be differentiable at $z_0$.

Here, $\Delta z$ is any complex number, so it may approach zero in a number of different directions. However, for this limit to exist, it must reach a unique limit f'($\Delta z$) independent of how $\Delta z$ approaches zero. For this reason, the complex conjugation $z \mapsto \bar z$ is nowhere differentiable. To see this consider the limit

$\lim_{w \to 0} \frac{ \overline{z + w}- \overline{z}}{w} = \lim_{w \to 0} \frac{ \overline{w}}{w} = \lim_{x, y \to 0} \frac{{x -y i}}{x + y i} \, .$

For $x=0$ this limit is $-1$ and for $y=0$ this limit is $1$, so there does not exist a unique complex limit.

Next