DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

Biholomorphic maps between complex domains

Definition

Let $U, V \subseteq C$ be complex domains (A complex domain is a nonempty connected open subset of $C$ ). A map $f : U \to V$ is biholomorphic if it is bijective (Injection, surjection, bijection), holomorphic, and its inverse $f^{- 1} : V \to U$ is holomorphic (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions). The map $f$ is then a biholomorphism from $U$ onto $V$ .

For local use, a holomorphic map is biholomorphic between neighbourhoods of $a$ and $f (a)$ when it restricts to a biholomorphism between complex domains $U^{'} \subseteq U$ and $V^{'} \subseteq V$ contained in those neighbourhoods and satisfying $a \in U^{'}$ and $f (a) \in V^{'}$ . The two membership conditions are what make the notion local at $a$ : without them $z \mapsto z^{2}$ would qualify at $0$ by restricting to a disc that avoids $0$ .

Biholomorphic maps between complex domains

Definition

Depends on

Dependency tree · 2 levels

Sources