The real exponential function and the number $e$ by a power series

Depends on

• A real power series about a centre, its interval of convergence, and its radius in $[0,+\infty]$
• The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
• Integer powers $a^m$
• The canonical natural $\iota(n) = n \cdot 1_F$ of a field
• Canonical naturals are positive and strictly increasing
• Series, partial sums, convergence and the sum, divergence, and the tail series

Used by

• Certified-action implementation of exponential Euler and ETD2 Algorithm
• Exp3 with importance-weighted loss feedback Algorithm
• Exp4 with unbiased estimated rewards Algorithm
• Exponential mechanism Algorithm
• Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules Algorithm
• Finite-class empirical-risk selection with exponential weights Algorithm
• Hedge Algorithm
• Randomized weighted majority with full information Algorithm
• Validation temperature scaling Algorithm
• Zero-shot prompt classification Algorithm
• A sample-size consequence for compression schemes Corollary
• Dobinski's formula expresses the Bell numbers as Bₙ=e⁻¹∑_ℓ≥0ℓⁿ/ℓ! Corollary
• Every continuous f with f(xy)=f(x)+f(y) is f(x)=c log x for a unique c, including c=0 Corollary
• Polynomial and exponential minimax estimation rates Corollary
• The elementary numerical bound 2<e<3 Corollary
• The exponential is positive and satisfies exp(-x)=1/exp(x) Corollary
• The number e is the unique x>0 satisfying ∫₁ˣdt/t=1 Corollary
• A bounded continuous normalized function that is not of positive type Counterexample
• Boundary-scale derivative blowup despite bounded ball data Counterexample
• Omitting the second-stage correction destroys cubic local accuracy Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Quadratic pure-birth rates can explode Counterexample
• The claimed univariate clean-quantile count exponent is too strong Counterexample
• Boys functions and lower incomplete-gamma notation Definition
• Cartesian Gaussian primitives, contractions, and integral conventions Definition
• Centered 1-sub-Gaussian reward Definition
• Diagonal Dirichlet sine-mode heat model with constant forcing Definition
• Given countable nonexplosive pure-jump chain with finite rates Definition
• Information units and logarithm-base convention Definition
• Kullback–Leibler divergence Definition
• Matrix multiplicative-weights game and update Definition
• Real powers for positive bases, with the zero-base positive-exponent convention Definition
• Robin constant and logarithmic capacity of a compact set Definition
• Sub-Gaussian variables and vectors, and sub-exponential variables Definition
• Symmetric image-text contrastive loss Definition
• The heat kernel on ℝⁿ and its causal extension Definition
• The moment generating function M_X(t)=E[e^tX] on a finite probability space Definition
• The scalar Poisson count law Definition
• The six hyperbolic functions and their natural domains Definition
• The standard flat function Definition

…and 167 more results.

Dependency tree · two levels

42 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

• MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
• J. Lebl, Basic Analysis, Logarithm and Exponential (standard reference, not scraped)