Differentiation of Complex Functions

Definition: Differentiation of Complex Functions

Let $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function on an open subset $\mathcal{D}$ and let $z_0\in \mathcal{D}$.

The derivative of $f$ at $z_0$ is the limit

$$\underset{z\to z_0}{\lim }\frac{f(z)-f(z_0)}{z-z_0}.$$

If it exists, we say that $f$ is differentiable at $z_0$.