The exponential is positive and satisfies $\exp (-x)=1\mathrm{/}\exp (x)$

Depends on

• The exponential addition formula $\exp(x+y)=\exp(x)\exp(y)$
• The real exponential function and the number $e$ by a power series
• Squares of nonzero elements are positive

Used by

• Dobinski's formula expresses the Bell numbers as Bₙ=e⁻¹∑_ℓ≥0ℓⁿ/ℓ! Corollary
• exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
• Lᵖ is uniformly convex for 1<p<∞ Corollary
• A discontinuous positive solution of F(x+y)=F(x)F(y) Counterexample
• A flat smooth real function has no holomorphic extension near zero Counterexample
• Smooth data do not force an analytic solution Counterexample
• Carleson tiles wave packets and tile order Definition
• The six hyperbolic functions and their natural domains Definition
• A family containing K₁ is viral for vacuous reasons Example
• A parameter ledger for the high-girth, high-chromatic alteration proof Example
• Fubini computes ∫₀¹∫₋₁¹ x exp(xy) dx dy by reversing the order Example
• The log-free product limit (1-2/n)ⁿ→exp(-2) Example
• The one-sided flat function is C^∞ with identically zero Taylor series Example
• FALSE: every smooth map between open subsets of the plane is real analytic False statement
• 1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp) Lemma
• Absolute real powers are Borel measurable and convex Lemma
• Chernoff bound for independent bernoulli trials Lemma
• The improper integral of e^-x² over ℝ is finite and positive Lemma
• A scalar first-order linear ODE has a unique solution given by the integrating-factor formula Theorem
• A tau-critical graph has no wide pure blockade with cograph pattern Theorem
• Addition formulas, identities, parity, and derivatives of the hyperbolic functions Theorem
• An α-narrow graph contains a perfect induced subgraph of order at least |V(G)|^1/α Theorem
• For all positive k,ℓ, some finite graph has girth greater than ℓ and chromatic number greater than k Theorem
• Gronwall's integral inequality with variable and constant coefficients Theorem
• Holder's inequality for finite sums and conjugate real exponents Theorem
• Minkowski's inequality for finite sums and real exponent p greater than one Theorem
• Morse stability with explicit parameter dependence Theorem
• Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm Theorem
• Regular normalized multiplicative Cauchy equations characterize the exponential Theorem
• The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents Theorem
• The exponential dominates every fixed nonnegative integer power at +∞ Theorem
• The exponential function is strictly increasing Theorem
• The exponential is the unique solution of y'=y with y(0)=1 Theorem
• The exponential tends to +∞ at +∞ and to 0 at -∞ Theorem
• Young's inequality for conjugate real exponents Theorem

Dependency tree · two levels

18 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)
• J. K. Hunter, An Introduction to Real Analysis, Chapter 10 (standard reference, not scraped)
• S. G. Johnson, Exponential Functions (standard reference, not scraped)