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Definition D1411
Analytic complex function

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

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

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

Let $U \subseteq \mathbb{C}$ be an D5008: Standard open complex set such that
 (i) $$U \neq \emptyset$$
A D4881: Complex function $f : U \to \mathbb{C}$ is analytic at $z_0 \in U$ if and only if $$\exists \, R > 0 \text{ and } a \in \mathbb{C}^{\mathbb{N}} : \forall \, z \in \mathbb{C} \left( |z - z_0| < R \quad \implies \quad f(z) = \sum_{n = 0}^{\infty} a_n (z - z_0)^n \right)$$
Children
 ▶ Analytic real function