The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$

Depends on

• The exponential series converges absolutely for every real argument
• Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius
• A power-series sum is infinitely differentiable inside its radius and satisfies $a_n=f^{(n)}(c)/\iota(n!)$ at its centre
• The factorial $n!$ and the falling factorial $n^{\underline{k}}$, defined by recursion in $\mathbb{N}$
• The canonical natural $\iota(n) = n \cdot 1_F$ of a field
• Canonical naturals are positive and strictly increasing

Used by

• A complete manifold with zero global injectivity radius Counterexample
• A flat smooth real function has no holomorphic extension near zero Counterexample
• Differentiation under an improper integral can fail without uniform domination Counterexample
• Hadamard instability despite analytic solvability Counterexample
• Infinite variance can defeat square-root-n CLT scaling Counterexample
• Smooth data do not force an analytic solution Counterexample
• The exponential is not uniformly continuous on ℝ Counterexample
• Two closed convex sets can have no strong separator Counterexample
• Carleson tiles wave packets and tile order Definition
• A nonzero smooth compactly supported bump Example
• A value outside the image can still be regular Example
• Cauchy law and its characteristic function Example
• Characteristic function of a gaussian law Example
• Conditional density of a bivariate normal law Example
• Fubini computes ∫₀¹∫₋₁¹ x exp(xy) dx dy by reversing the order Example
• Geodesics in the Poincare upper half-plane Example
• Independent sums via characteristic functions Example
• Polynomial Gaussians are Schwartz Example
• The additive and multiplicative real Lie groups Example
• The one-sided flat function is C^∞ with identically zero Taylor series Example
• Two equations implicitly determine two variables near the origin Example
• A regular value need not belong to the image False statement
• 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
• Characteristic function of a normal law Lemma
• Chernoff bound for independent bernoulli trials Lemma
• Gaussian even moments for Brownian increments Lemma
• The plane Gaussian integral equals π by polar coordinates Lemma
• The standard normal density has total mass one Lemma
• Uniform sine integral bound and dirichlet value Lemma
• A scalar first-order linear ODE has a unique solution given by the integrating-factor formula Theorem
• Addition formulas, identities, parity, and derivatives of the hyperbolic functions Theorem
• Continuity and derivatives of positive-base real powers Theorem
• Gronwall's integral inequality with variable and constant coefficients Theorem
• Landau's root limit: log x is the limit of 2ⁿ times (x^(1/2ⁿ) minus 1) Theorem
• Regular normalized multiplicative Cauchy equations characterize the exponential Theorem
• The complex exponential is entire and its complex derivative is itself Theorem
• The exponential function is strictly increasing Theorem
• The exponential is the unique solution of y'=y with y(0)=1 Theorem
• The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t Theorem

…and 3 more results.

Dependency tree · two levels

33 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)
• J. Lebl, Basic Analysis, Analytic Functions (standard reference, not scraped)