DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Admissible cycles for the residue theorem

Let $Ω \subseteq C$ be open, let $f$ be meromorphic on $Ω$ in the sense of Meromorphic functions on a plane domain, and let $S \subseteq Ω$ be its pole set. By Poles of a meromorphic function form a closed discrete set and are at most countable, $S$ is closed and discrete in $Ω$ .

A complex cycle $Γ$ is admissible for the residue theorem in $Ω$ when

$Γ^{*} \subseteq Ω ∖ S$ ,
$Γ$ is null-homologous in $Ω$ in the sense of Null-homologous cycles and homologous cycles in an open set.

For such a pair $(Γ, f)$ the candidate residue sum is

$\sum_{a \in S} n (Γ, a) Res (f, a) .$

Only points with nonzero index can contribute, and the next lemma shows that there are only finitely many of them.

Depends on