ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

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

Example

For every integer $m \geq 1$ ,

$f (z) = z^{m}$

has a pole of order $m$ at $\infty$ , while

$g (z) = \frac{1}{z^{m}}$

has a removable singularity at $\infty$ .

Facts & Assumptions

Given: An integer $m \geq 1$ .

[L1]

A singularity at infinity is classified by the singularity of the pulled-back function $h (w) = f (1 / w)$ at $w = 0$ (Isolated singularities at infinity).

[L2]

A pole is characterized by a finite nonzero principal part, and a removable singularity by vanishing principal part or holomorphic extendability (Characterizations of poles, Characterizations of removable singularities).

Verification

technique · direct

1.1L1L2

For $f (z) = z^{m}$ , the pulled-back function is $f (1 / w) = w^{- m}$ on $0 < ∣ w ∣ < 1 / R$ , so it has a pole of order $m$ at $0$ by [L2]; therefore $f$ has a pole of order $m$ at $\infty$ by [L1].

2.1L1L2∎

For $g (z) = z^{- m}$ , the pulled-back function is $g (1 / w) = w^{m}$ , which is holomorphic at $0$ and vanishes there; by [L2] the singularity at $0$ is removable, so $g$ has a removable singularity at $\infty$ by [L1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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.