Complex differentiability

Definition

Let $U \subseteq \mathbb{C}$ be an open set, let $f : U \to \mathbb{C}$ be a function, and let $z_0 \in U$. The function $f$ is complex differentiable at $z_0$ if the limit $$f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$$ exists in $\mathbb{C}$. Here $h$ runs through the nonzero complex numbers with $z_0 + h \in U$. The number $f'(z_0)$ is the complex derivative of $f$ at $z_0$. The limit means: for each $\varepsilon > 0$ there is $\delta > 0$ such that $\left| \frac{f(z_0 + h) - f(z_0)}{h} - f'(z_0) \right| < \varepsilon$ for all such $h$ with $0 < |h| < \delta$.