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

Isolated singularities at infinity

Definition

Let $f$ be holomorphic on an exterior region $∣ z ∣ > R$ . Write

$g (w) : = f (1 / w)$

for $0 < ∣ w ∣ < 1 / R$ . Then $f$ is said to have an isolated singularity at infinity when $g$ has an isolated singularity at $0$ .

The singularity at infinity is:

removable when $g$ is removable at $0$ ;
a pole of order $m$ when $g$ has a pole of order $m$ at $0$ ;
essential when $g$ is essential at $0$ .

Remarks

This is the singularity-type dictionary at infinity only. The later residue theorem page introduces the separate residue-at-infinity convention used in global contour formulas.

Used by

Positive powers have poles at infinity and their reciprocals have removable singularities there Example

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Jean-Baptiste Campesato, MAT334 course page and notes index (standard reference, not scraped)
L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 5 §1.3 (standard reference, not scraped)