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
\[ \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}. \]
If it exists, we say that \(f\) is differentiable at \(z_0\).