Change of variable in an improper integral

Depends on

• Monotone change of variable for Riemann-integrable functions
• Improper integrals over unbounded intervals
• Improper integrals at a finite singular endpoint
• Improper integrals with several singular ends
• Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$, with the dictionary to monotone sequences
• The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
• The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$

Used by

• The closed form for the volume of the unit n-ball Corollary
• Γ(1/2)=√π from the Gaussian integral Corollary
• Differentiation under an improper integral can fail without uniform domination Counterexample
• Iid strong law fails at infinite absolute mean Counterexample
• The complementarity product needs extra regularity Counterexample
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• A Caputo startup singularity and a graded-grid calculation Example
• An exact Gaussian Laplace integral on the noncompact line Example
• The positive-type Gaussian on the real line and its cyclic model Example
• The scaled Gaussian integral and its parameter derivative Example
• The substitution x=1/t exchanges the two rational p-tests Example
• Vanishing viscosity selects the Hopf--Lax solution for bounded data Example
• A deterministic-prefix envelope sum for the sharper UCB count bound Lemma
• A normal variable has finite fourth moment Lemma
• A one-coordinate projection risk bound Lemma
• Cartesian Gaussian moments and exponent-parameter derivatives Lemma
• Gaussian integral for a positive-definite quadratic form Lemma
• Holomorphic Morse normalization and Gaussian contour Lemma
• Integral of the fractional power kernel Lemma
• Integral representations and a range bound for integer Bessel functions Lemma
• Moment recurrence for the scaled complementary error function Lemma
• Parallel Gaussian information density obeys a finite-variance law of large numbers Lemma
• Semigroup law for fractional integrals Lemma
• Signed finite remainder for the erfcx asymptotic expansion Lemma
• Standard-normal CDF, tail, and quantile identities for power calculations Lemma
• The Hilbert transform is the tempered convolution with pv(1/(pi x)) and has signum Fourier multiplier Lemma
• The improper integral of e^-x² over ℝ is finite and positive Lemma
• The sine integral under Countable Choice: uniform bounds and the value pi/2 Lemma
• The standard normal CDF has unique symmetric quantiles Lemma
• Weierstrass log-Gamma series converges to a Gamma branch Lemma
• Weighted monomial integrals and monomial norms for the disc, ball and polydisc Lemma
• Symmetry and the trigonometric form of the real Beta integral Proposition
• Boys integral, power series, recurrence, and incomplete-gamma representation Theorem
• Hermite recurrence formulas for Gaussian nuclear-attraction and two-electron Coulomb integrals Theorem
• Khintchine's inequality for finite Rademacher sums Theorem
• Positive Coulomb bilinear form, Cauchy–Schwarz, and screening bound Theorem
• The real Gamma function is smooth and its derivatives are logarithmic moments Theorem

Dependency tree · two levels

32 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

• William F. Trench, Introduction to Real Analysis, Theorem 3.4.13 (standard reference, not scraped)