Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13
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 integral exponential as the inverse of
Definition
Because
is a bijection ( is a continuous strictly increasing bijection), it has an inverse function in the sense of Injection, surjection, bijection. Define the integral exponential
Thus, for and ,
Depends on
Used by
- Every continuous f with f(xy)=f(x)+f(y) is f(x)=c log x for a unique c, including c=0 Corollary
- The integral logarithm L is the published natural logarithm Corollary
- The inverse E satisfies E(a+b)=E(a)E(b) Corollary
- The number e is the unique x>0 satisfying ∫₁ˣdt/t=1 Corollary
- log is the unique continuous f:(0,∞)→ℝ with f(xy)=f(x)+f(y) and f(e)=1 Theorem
- The inverse E is differentiable, E'=E, and E(0)=1 Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 35 results over 10 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
- OpenStax, Calculus Volume 1, Section 6.7 (standard reference, not scraped)