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Definition D3711
Complex power series

Let $\mathbb{C}^{\mathbb{C}}$ be the D3093: Set of functions from $\mathbb{C}$ to $\mathbb{C}$.
Let $r : \mathbb{N} \to \mathbb{C}$ be a D339: Complex sequence.
The complex power series with respect to $r$ and $z_0 \in \mathbb{C}$ is the D62: Sequence $$\mathbb{N} \to \mathbb{C}^{\mathbb{C}}, \quad N \mapsto \left( z \mapsto \sum_{n = 0}^N r_n (z - z_0)^n \right)$$

Let $\mathbb{C}^{\mathbb{C}}$ be the D3093: Set of functions from $\mathbb{C}$ to $\mathbb{C}$.
Let $r : \mathbb{N} \to \mathbb{C}$ be a D339: Complex sequence.
The complex power series with respect to $r$ and $z_0 \in \mathbb{C}$ is the D62: Sequence $$\mathbb{N} \to \mathbb{C}^{\mathbb{C}}, \quad N \mapsto \left( f_N : \mathbb{C} \to \mathbb{C}, \quad f_N(z) = \sum_{n = 0}^N r_n (z - z_0)^n \right)$$
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 ▶ Analytic complex function