Alphabeta MathThe Integral Logarithm and the Equivalence of Its Characterisations: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Equivalent Forms of Completeness
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Integral Logarithm and the Equivalence of Its Characterisations
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
log 2 from four characterisations: hyperbolic area, the alternating harmonic series, Landau iterates and exp⁻¹(2)
Example
Dropping f(e)=1 leaves the whole family c log, including logarithms to other bases and the zero function
Counterexample
Assuming choice, a Hamel-basis additive map transported through exp gives a discontinuous logarithmic function that is not c log
Counterexample
For every x>2 the uncontinued Mercator series at u=x-1 diverges
Example