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.
Sine and cosine defined by their real power series
Definition
For , define
These are real power series in the sense of A real power series about a centre, its interval of convergence, and its radius in . Their convergence for every real argument is discharged by The sine and cosine power series converge absolutely for every real argument ↗.
Depends on
Used by
- The classical Weierstrass function Definition
- The Bartle-Sherbert bounds 2.828 < pi < 3.185 Example
- Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3 Lemma
- The sine and cosine power series converge absolutely for every real argument Lemma
- Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ Theorem
- Standard Maclaurin expansions Theorem
- The derivatives of sine and cosine are cosine and minus sine Theorem
Dependency tree · two levels
7 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, Chapter 4 (standard reference, not scraped)
- C. Schmeiser, Introduction to Analysis (standard reference, not scraped)