DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Holomorphic germs at a point

Definition

Fix a point $a \in C$ . Two holomorphic functions $f : U \to C$ and $g : V \to C$ , defined on open neighbourhoods $U, V$ of $a$ (Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions), are equivalent at $a$ when there is an open neighbourhood $W$ of $a$ with $W \subseteq U \cap V$ and

$f ∣_{W} = g ∣_{W} .$

This is an equivalence relation. An equivalence class is a holomorphic germ at $a$ and is written $[f]_{a}$ when $f$ is one of its representatives.

If $[f]_{a} = [g]_{a}$ , then $f (a) = g (a)$ because every witnessing neighbourhood contains $a$ . So the value of a germ at its base point is well defined and may be written $[f]_{a} (a)$ or simply $f (a)$ when no confusion can occur.

Depends on

Complex differentiability at a point, the complex derivative, holomorphic functions, and entire functions

Used by

Function elements and direct analytic continuation Definition
The complete analytic function generated by one germ Definition
Holomorphic germs at a point form a local ring Theorem
The terminal germ of a continuation along a fixed path is chain-independent Theorem

Dependency tree · two levels

4 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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 8 §1.2 (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Ch. 4 (standard reference, not scraped)