The exponential tends to $+\mathrm{\infty }$ at $+\mathrm{\infty }$ and to $0$ at $-\mathrm{\infty }$

Depends on

• The real exponential function and the number $e$ by a power series
• A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum
• The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$
• Limits at $+\infty$ and $-\infty$, and infinite limits at a point

Used by

• Finite truncated tanh–sinh, exp–sinh, and sinh–sinh trapezoid rules Algorithm
• The exponential is a continuous bijection from ℝ onto (0,∞) Corollary
• A complete manifold with zero global injectivity radius Counterexample
• Differentiation under an improper integral can fail without uniform domination Counterexample
• Knapp rules out extension below the Tomas exponent Counterexample
• Misfit evidence and the potential posterior law Definition
• Quasisymmetric homeomorphisms of the line and circle Definition
• Backward Euler and BDF2 damp a fast scalar mode while trapezoidal does not Example
• Complex sine is unbounded on the imaginary axis Example
• Exponential singular decay, logarithmic balls, and spectral gaps Example
• Exponential singular-value decay and an exponentially weighted truth class Example
• Polynomial singular-value decay and a power Gaussian prior Example
• Power maps, endpoint distortion, and a non-Möbius quasisymmetric circle map Example
• The Gaussian integral on the line with a certified sinh-sinh truncation tail Example
• The positive-type Gaussian on the real line and its cyclic model Example
• A deterministic-prefix envelope sum for the sharper UCB count bound Lemma
• A double-exponential envelope bounds the omitted trapezoid tails Lemma
• A sufficiently high power regularizes finite logarithmic endpoint factors Lemma
• Fresnel whole-line moment from the damped complex Gaussian Lemma
• Maximum principle on a closed strip for bounded holomorphic functions Lemma
• Stirling's factorial asymptotic holds up to a positive constant Lemma
• The finite Rényi generators are convex with stated boundary values Lemma
• The improper integral of a power at zero Lemma
• The plane Gaussian integral equals π by polar coordinates Lemma
• Addition formulas, identities, parity, and derivatives of the hyperbolic functions Theorem
• An analytic-strip error bound for the infinite trapezoid sum Theorem
• An entire function of polynomial growth is a polynomial Theorem
• Complex sine and cosine are unbounded on the complex plane Theorem
• Empirical-Bernstein confidence sequence under adaptive sampling Theorem
• Equal-spherical mixture assignments and objective converge to k-means Theorem
• Existence of a Dirichlet process by normalized gamma Poisson random measure Theorem
• Exponential and Weibull survival specifications define probability laws Theorem
• Interior finite-temperature stationarity gives Fermi-Dirac occupations Theorem
• Knapp necessary condition for spherical L2 restriction Theorem
• The double-exponential maps and their change-of-variable identities Theorem
• The logarithm grows more slowly than every positive real power Theorem
• The p-series for a real exponent p converges exactly when p is greater than one Theorem

Dependency tree · two levels

31 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)