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 boundary mode has a different concentration rate Counterexample
• 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
• Omitting the second-stage correction destroys cubic local accuracy Counterexample
• Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
• Smooth data do not force an analytic solution Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• The exponential is not uniformly continuous on ℝ Counterexample
• The final-time face is not part of the parabolic boundary Counterexample
• Two closed convex sets can have no strong separator Counterexample
• Carleson tiles wave packets and tile order Definition
• Cartesian Gaussian primitives, contractions, and integral conventions Definition
• Conjugate-form prior for an exponential family Definition
• Hermite Gaussian and Boys Coulomb auxiliary functions Definition
• The heat kernel on ℝⁿ and its causal extension Definition
• A consistent but unstable explicit spectral-grid heat step Example
• A diagonal stiff diffusion step with linear reaction Example
• A finite Euler adjoint gradient can differ from the exact flow gradient Example
• A Gaussian maximum without independence Example
• A Hopf--Lax solution with a forming corner from smooth data Example
• A mean-matched count law can be overdispersed relative to Poisson Example
• A nonnormal action grows transiently while its Arnoldi residual certificate remains valid Example
• A nonzero smooth compactly supported bump Example
• A one-dimensional Arnoldi projection misses a slow mode of a stiff diagonal matrix Example
• A one-observation Bernoulli-logit Fisher-scoring step Example
• A one-observation Poisson-log Fisher-scoring step Example
• A regularized logistic finite sum with an exact minimizer Example
• A second-order Taylor polynomial computed from gradient and Hessian data Example
• A smooth scattered surface inherits the Matérn fill-distance rate Example
• A stable consistent explicit spectral-grid heat step Example
• A stable coupled model has an unstable explicit fast microstep Example
• A stiff penalty equation defeats a fixed explicit step Example
• A two-interval shooting residual has a block-bidiagonal sensitivity Jacobian Example
• A two-state witness to short-run contrastive-divergence bias Example
• A value outside the image can still be regular Example
• A wide tanh network stays lazy on a fixed time horizon Example
• An explicitly treated stiff decay mode is unstable for a large step Example

…and 135 more results.

Dependency tree · two levels

35 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)