DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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 $0$ .

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, §5.6 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, §1.2 (standard reference, not scraped)