Alphabeta MathHow 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.
This page keeps Cauchy principal values distinct from genuine improper convergence
Remark
Cauchy principal values at a finite singularity and on the real line and Conventions and proved scope for improper integrals already distinguish two notions: an improper integral on the real line exists only when the one-sided tails or endpoint pieces converge separately, while a principal value couples symmetric truncations and can exist even when the separate one-sided limits diverge.
This page keeps that distinction rigid. A contour with a real-axis indentation usually computes a principal value first. It gives an ordinary improper value only when the real integral has already been shown to converge by an independent tail or endpoint argument.
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Used by
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Sources
- L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §5.3 (standard reference, not scraped)