ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation F10571 on D1411: Analytic complex function
F10571
Formulation 3
Let $\mathbb{C}$ be the D1378: Standard complex metric space such that
(i) $E \subseteq \mathbb{C}$ is a D78: Subset of $\mathbb{C}$
(ii) \begin{equation} E \neq \emptyset \end{equation}
(iii) $z_0 \in E$ is an D1387: Interior point of $E$ in $\mathbb{C}$
A D4881: Complex function $f : E \to \mathbb{C}$ is analytic at $z_0$ if and only if \begin{equation} \exists \, R > 0, a \in \mathbb{C}^{\mathbb{N}} : \forall \, z \in B(z_0, R) : f(z) = a_0 (z - z_0)^0 + a_1 (z - z_0)^1 + a_2 (z - z_0)^2 + \cdots \end{equation}