DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-02

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.

The six hyperbolic functions and their natural domains

Definition

For $x \in R$ , define $\sinh x : = \frac{\exp x - \exp (- x)}{2}, \cosh x : = \frac{\exp x + \exp (- x)}{2},$ $\tanh x : = \frac{\sinh x}{\cosh x}, sech x : = \frac{1}{\cosh x} .$ For $x \neq 0$ , define $\coth x : = \frac{\cosh x}{\sinh x}, csch x : = \frac{1}{\sinh x} .$

The positivity of $\exp$ makes $\cosh x > 0$ for every $x$ . If $x \neq 0$ , then $x \neq - x$ , so strict increase of $\exp$ makes $\exp x \neq \exp (- x)$ ; hence $\sinh x \neq 0$ . Thus all four quotients above are defined on exactly their stated domains (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential function is strictly increasing).

Depends on

Used by

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)