Definition of Complex Differentiation

Let $f:D\to \mathbb{C}$ be a complex function defined on an open set $D\subset \mathbb{C}$. $f$ is said to be differentiable at $z_0\in D$ if…

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

exists. This limit is denoted by $f^{\prime }(z_0)$ or by $\left(\frac{df}{dz}\right)z_0$.