Alphabeta Math
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.

The residue of an isolated singularity

Definition

Let f have an isolated singularity at a. By Laurent expansion on an annulus, on every sufficiently small punctured disc about a it has a Laurent expansion

f(z)=∑n∈Zcn(z−a)n.

By Laurent coefficients are given by contour integrals and are unique, the coefficient c−1 is uniquely determined on that disc, and Laurent coefficients are independent of the intermediate radius shows that shrinking the disc does not change it. It is the residue of f at a, written

Res⁡(f,a):=c−1.

Remarks

The residue depends only on the singularity of f at a, not on which small punctured disc is used to write the Laurent series.

Depends on

Dependency tree · 2 levels

Sources