AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generated

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.

Exact recurrence for polynomial Fourier moments

Definition

Input real endpoints $a < b$ , a real frequency $ω \neq 0$ , and an integer $m \geq 0$ . Set $μ_{0} = (e^{i ω b} - e^{i ω a}) / (i ω) .$ For $k = 1, \dots, m$ in increasing order, set $μ_{k} = (b^{k} e^{i ω b} - a^{k} e^{i ω a}) / (i ω) - k μ_{k - 1} / (i ω) .$ Return the complex list $(μ_{0}, \dots, μ_{m})$ . The divisions are defined by $ω \neq 0$ . The procedure works in exact real and complex arithmetic; its exact moment identities are stated and proved separately on this page.

Depends on

Finite-interval oscillatory integral and phase convention

Used by

Dependency tree · two levels

5 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

A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, §3.2 (standard reference, not scraped)
D. Levin, Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, §1 (standard reference, not scraped)