The exponential dominates every fixed nonnegative integer power at $+\mathrm{\infty }$

Depends on

• The real exponential function and the number $e$ by a power series
• The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
• Limits at $+\infty$ and $-\infty$, and infinite limits at a point
• 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
• Laws of finite sums and finite products

Used by

• A flat smooth real function has no holomorphic extension near zero Counterexample
• A smooth flat refinement has a strict minimum on every line through the origin but no local extremum Counterexample
• Hadamard instability despite analytic solvability Counterexample
• Smooth data do not force an analytic solution Counterexample
• The exponential is not uniformly continuous on ℝ Counterexample
• Carleson tiles wave packets and tile order Definition
• Concrete logarithmic, polynomial, and exponential growth comparisons Example
• Polynomial Gaussians are Schwartz Example
• The one-sided flat function is C^∞ with identically zero Taylor series Example
• The scaled Gaussian integral and its parameter derivative Example
• FALSE: every smooth map between open subsets of the plane is real analytic False statement
• FALSE: existence of partial derivatives satisfying Cauchy–Riemann everywhere on an open set implies holomorphy False statement
• Euclidean Gaussian transform with the 2π normalization Lemma
• Explicit compactly supported smooth cutoffs Lemma
• Exponential decay dominates every inverse power near zero Lemma
• The exponential roadmap and its circularity hazards Remark
• Euler's Gamma integral converges exactly for positive real parameters Theorem
• Every finite family with the Erdős–Hajnal property is viral Theorem
• The logarithm grows more slowly than every positive real power Theorem
• The real Gamma function is smooth and its derivatives are logarithmic moments Theorem
• The real Gamma functional equation Γ(s+1)=sΓ(s) Theorem

Dependency tree · two levels

43 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)
• MIT 18.102, Chapter 4 notes (standard reference, not scraped)