DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Meromorphic functions on a plane domain

Definition

Let $Ω \subseteq C$ be a nonempty connected open set. A function $f : Ω ∖ P \to C$ , where $P \subseteq Ω$ , is meromorphic on $Ω$ when

$f$ is holomorphic on $Ω ∖ P$ , and
every point of $P$ is a pole of $f$ in the sense of Isolated singularities: removable, poles, and essential singularities.

The set $P$ is the pole set of the meromorphic function.

Remarks

If $P = \emptyset$ , the function is simply holomorphic on $Ω$ .

This definition is deliberately local. The later page on the argument principle adds the quotient and divisor viewpoints, but this page works only with the isolated-pole description.

Depends on

Isolated singularities: removable, poles, and essential singularities

Meromorphic functions on a plane domain

Definition

Remarks

Depends on

Dependency tree · 1 level

Sources