Solution
= Solution
A function is <holomorphic> on an <open set> when it is <complex differentiable> at every point of that set. Explicitly, for every $z\in U$, the limit
$$
\boxed{f'(z)=\lim_{h\to0,\ h\in\mathbb C\setminus\{0\}}\frac{f(z+h)-f(z)}h}
$$
exists as a finite <complex number>. The limit is independent of the direction in which the complex increment approaches zero; existence of <derivatives> only along the real axis would not suffice.