Analytic function

Let \(\Omega\subseteq\CC\) be open, let \(f\colon\Omega\to\CC\), and let \(z_0\in\Omega\). The function \(f\) is analytic at \(z_0\) if there exist a power series \(\sum_{n\ge0}a_n(z-z_0)^n\) with radius of convergence \(R>0\) and an open neighborhood \(U\subseteq\Omega\) of \(z_0\) with \(\abs{z-z_0}<R\) for all \(z\in U\), such that \[ f(z)=\sum_{n\ge0}a_n(z-z_0)^n\qquad\text{for all } z\in U. \] The function \(f\) is analytic on \(\Omega\) if it is analytic at every point of \(\Omega\).