DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Complex analytic functions as locally representable by convergent power series

Definition

Let $U \subseteq C$ be open and let $f : U \to C$ . The function $f$ is analytic at $a \in U$ if there are $r > 0$ and complex coefficients $(c_{n})_{n \geq 0}$ such that $B (a, r) \subseteq U$ and $f (z) = \sum_{n = 0}^{\infty} c_{n} (z - a)^{n} (z \in B (a, r)),$ with convergence in the sense of Complex series, absolute convergence, complex power series, and radius of convergence. It is analytic on $U$ if it is analytic at every point of $U$ .

This terminology is distinct from holomorphic in Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions: analytic is defined by local power-series representation, while holomorphic is defined by complex differentiability.

Complex analytic functions as locally representable by convergent power series

Definition

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