Alphabeta MathDefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 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.
Real powers for positive bases, with the zero-base positive-exponent convention
Definition
For and , define For the supplementary zero-base convention, define when .
The expressions and for are left undefined. Negative bases are not assigned arbitrary real powers.
Depends on
Used by
- xˣ tends to one as x tends to zero from the right Example
- Change of base and inversion of the positive-base real exponential Theorem
- Continuity and derivatives of positive-base real powers Theorem
- Holder's inequality for finite sums and conjugate real exponents Theorem
- Landau's root limit: log x is the limit of 2ⁿ times (x^(1/2ⁿ) minus 1) Theorem
- Minkowski's inequality for finite sums and real exponent p greater than one Theorem
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents Theorem
- The exponential definition of real powers agrees with the existing rational powers Theorem
- The logarithm grows more slowly than every positive real power Theorem
- The rational-supremum construction of real powers agrees with the exponential construction Theorem
- The weighted arithmetic-geometric mean inequality for real weights Theorem
- Young's inequality for conjugate real exponents Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 47 results over 13 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
- 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)