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.
The six hyperbolic functions and their natural domains
Definition
For , define For , define
The positivity of makes for every . If , then , so strict increase of makes ; hence . Thus all four quotients above are defined on exactly their stated domains (The exponential is positive and satisfies , The exponential function is strictly increasing).
Depends on
Used by
- Geodesics in the Poincare upper half-plane Example
- The Gaussian integral on the line with a certified sinh-sinh truncation tail Example
- Volume growth in euclidean and hyperbolic space Example
- A one-coordinate projection risk bound Lemma
- A truncated covering integral bounds empirical Rademacher complexity Lemma
- Clean Gaussian covariance has a capped squared operator lower rate Lemma
- Sparse Gaussian scalar certificate and extremal profile Lemma
- Uniform deviation for squared-projection slabs Lemma
- Model functions solve the constant curvature jacobi equation Proposition
- Upper half-space model geometry Proposition
- Addition formulas, identities, parity, and derivatives of the hyperbolic functions Theorem
- Khintchine's inequality for finite Rademacher sums Theorem
- Logarithm formulas for inverse sinh, inverse cosh, and inverse tanh on their natural domains Theorem
Dependency tree · two levels
15 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
- J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)