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 $$\begin{equation} \exists \, R > 0, a \in \mathbb{C}^{\mathbb{N}} : \forall \, z \in B(z_0, R) : f(z) = \sum_{n = 0}^{\infty} a_n (z - z_0)^n \end{equation}$$

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 $$\begin{equation} \exists \, R > 0, a \in \mathbb{C}^{\mathbb{N}} : \forall \, z \in B(z_0, R) : f(z) = \lim_{N \to \infty} \sum_{n = 0}^N a_n (z - z_0)^n \end{equation}$$

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 $$\begin{equation} \exists \, R > 0, a \in \mathbb{C}^{\mathbb{N}} : \forall \, z \in B(z_0, R) : f(z) = a_0 (z - z_0)^0 + a_1 (z - z_0)^1 + a_2 (z - z_0)^2 + \cdots \end{equation}$$

Let $U \subseteq \mathbb{C}$ be an D5008: Standard open complex set such that

(i)	$$\begin{equation} U \neq \emptyset \end{equation}$$

A D4881: Complex function $f : U \to \mathbb{C}$ is analytic at $z_0 \in U$ if and only if $$\begin{equation} \exists \, R > 0 \text{ and } a \in \mathbb{C}^{\mathbb{N}} : \forall \, z \in \mathbb{C} \left( |z - z_0| < R \quad \implies \quad f(z) = \sum_{n = 0}^{\infty} a_n (z - z_0)^n \right) \end{equation}$$