Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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  • 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.
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The one-variable function 1 / z does not extend across the origin

Statement refuted

Refuted claim: the one-variable function 1/z extends holomorphically across 0.

Facts & Assumptions

Given: The holomorphic function f(z)=1/z on 0<∣z∣<1.

[L1]

A simple pole is a pole of order 1 (Simple poles).

[L2]

A pole is exactly a nonremovable singularity whose reciprocal has a zero at the singular point (Characterizations of poles).

Counterexample

technique · direct
1.1

The function f(z)=1/z satisfies z f(z)=1, so 0 is a pole of order 1, that is, a simple pole, by [L1].

givenL1algebra
2.1

By [L2], a pole is not removable. Hence 1/z does not extend holomorphically across 0. This is the one-variable contrast to the several-variable puncture theorem.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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.

Sources