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 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
- The derivatives of sine and cosine are cosine and minus sine Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 34 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)