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.

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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.

Depends on

Used by

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Sources

L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)
E. Stein and R. Shakarchi, Complex Analysis, Ch. 1 (standard reference, not scraped)