DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Symmetric Cauchy principal value by subtraction

Definition

Let $a < c < b$ and $f \in C^{1} ([a, b])$ . The symmetric Cauchy principal value of $f (x) / (x - c)$ , when the limit exists, is $PV ⁣ \int_{a}^{b} \frac{f (x)}{x - c} d x : = \lim_{ε ↓ 0} (\int_{a}^{c - ε} \frac{f (x)}{x - c} d x + \int_{c + ε}^{b} \frac{f (x)}{x - c} d x),$ where $0 < ε < \min (c - a, b - c)$ . The first divided difference of a C1 function extends continuously supplies the continuous subtraction remainder; the existence and value of the limit are proved in The Cauchy principal-value subtraction identity ↗.

Depends on

Used by

Dependency tree · two levels

8 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

Lloyd N. Trefethen and J. A. C. Weideman, The Exponentially Convergent Trapezoidal Rule, SIAM Review 56(3), 2014 (standard reference, not scraped)