AlgorithmAlgorithm: 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.

Bounded integration-by-parts expansion for an oscillatory moment

Definition

Input $a < b$ , $ω \in R ∖ {0}$ , a real phase $g \in C^{r + 1} [a, b]$ with $∣ g^{'} ∣ \geq c > 0$ , a complex amplitude $p \in C^{r} [a, b]$ , an integer $r \geq 1$ , and a certified real bound $M \geq ∥ T_{g}^{r} p ∥_{\infty}$ . Compute $S_{r} (p) = \sum_{j = 0}^{r - 1} \frac{(- 1)^{j}}{(i ω)^{j + 1}} {[\frac{T_{g}^{j} p (x)}{g^{'} (x)} e^{i ω g (x)}]}_{a}^{b}, ϵ_{r} = \frac{(b - a) M}{∣ ω ∣^{r}} .$ Return $(S_{r} (p), ϵ_{r})$ . The nonstationary recurrence and the need for the certified derivative bound are specified in Repeated integration by parts for a nonstationary oscillatory integral. If exact phase moments are available, they may instead be passed as exact moment data to a Filon rule; this algorithm makes no general closed-form claim for nonlinear phases.

Depends on

Used by

Filon quadrature from nodal interpolation and phase moments Algorithm

Dependency tree · two levels

11 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

NIST Digital Library of Mathematical Functions, §2.3(i) (standard reference, not scraped)
A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, §2 (standard reference, not scraped)