A holomorphic function equals its Taylor series at $p$ on every open disc centred at $p$ whose closure lies in the domain.Suggest a slogan
Let \(\Omega\subseteq\CC\) be open, let \(f\) be holomorphic on \(\Omega\), and let \(D\) be an open disc centred at \(p\) whose closure is contained in \(\Omega\). Then for every \(z\in D\) the series below converges and \[ f(z)=\sum_{k\geq0}c_k(z-p)^k,\qquad c_k=\frac{f^{(k)}(p)}{k!}. \] In particular, \(f\) is analytic at every point of \(\Omega\).
See [@SS03]. Conversely, a function analytic on \(\Omega\) is holomorphic on \(\Omega\), since a power series is complex differentiable inside its disc of convergence; so holomorphic and analytic functions on \(\Omega\) coincide.