• the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition...
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  • framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be...
    11 KB (1,624 words) - 06:45, 6 February 2024
  • Thumbnail for Thomas Joannes Stieltjes
    Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral. Stieltjes was...
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  • Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced [dɑ̃ʒwa]), Luzin integral or Perron...
    18 KB (2,872 words) - 11:57, 31 March 2025
  • Thumbnail for Integral
    Lebesgue–Stieltjes integral, further developed by Johann Radon, which generalizes both the Riemann–Stieltjes and Lebesgue integrals. The Daniell integral, which...
    69 KB (9,288 words) - 03:06, 30 June 2025
  • products. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be...
    10 KB (1,336 words) - 21:43, 12 March 2025
  • Thumbnail for Itô calculus
    Itô calculus (redirect from Itô integral)
    central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators...
    31 KB (4,554 words) - 03:50, 6 May 2025
  • The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform...
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  • Thumbnail for Riemann integral
    Riemann–Stieltjes integral, and most disappear with the Lebesgue integral, though the latter does not have a satisfactory treatment of improper integrals. The...
    42 KB (5,479 words) - 01:14, 12 April 2025
  • Thumbnail for Fourier transform
    \xi }\,d\mu ,} and called the Fourier-Stieltjes transform due to its connection with the Riemann-Stieltjes integral representation of (Radon) measures....
    177 KB (21,313 words) - 19:14, 8 July 2025
  • Riemann–Stieltjes integral). The circle ∘ {\displaystyle \circ } is a notational device, used to distinguish this integral from the Itô integral. Many integration...
    10 KB (1,777 words) - 13:55, 1 July 2025
  • f^{-1}} is not differentiable: it suffices, for example, to use the Stieltjes integral in the previous argument. On the other hand, even though general monotonic...
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  • : 13–15  Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered. The first integral, with broad...
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  • Integration by parts (category Integral calculus)
    formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation...
    37 KB (7,124 words) - 18:34, 21 June 2025
  • Length Area Volume Probability Moving average Riemann sum Riemann–Stieltjes integral Bounded variation Jordan content Cauchy principal value Measure (mathematics)...
    2 KB (221 words) - 02:51, 2 May 2022
  • Riemann–Stieltjes integral, along with an appropriate function of bounded variation, gives a definition of integral equivalent to the Lebesgue–Stieltjes integral...
    11 KB (1,647 words) - 14:36, 23 July 2024
  • Thumbnail for Mertens conjecture
    It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)), and again in print by Franz Mertens (1897)...
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  • integrator in a Stieltjes integral is represented as the differential of a function. Formally, the differential appearing under the integral behaves exactly...
    27 KB (3,994 words) - 18:39, 27 May 2025
  • Kolmogorov integral (or Kolmogoroff integral) is a generalized integral introduced by Kolmogoroff (1930) including the Lebesgue–Stieltjes integral, the Burkill...
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  • bounded, continuous function g: R → R, where the integrals involved are Riemann–Stieltjes integrals. Note that if X and X1, X2, ... are random variables...
    2 KB (287 words) - 06:33, 14 April 2025
  • Thumbnail for Lebesgue integral
    Measure Sigma-algebra Lebesgue space Lebesgue–Stieltjes integration Riemann integral Henstock–Kurzweil integral This approach can be found in most treatments...
    41 KB (5,918 words) - 20:43, 16 May 2025
  • Thumbnail for Bernhard Riemann
    integral in his habilitation. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral...
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  • variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions. Another characterization states that...
    61 KB (8,441 words) - 20:55, 29 April 2025
  • the Darboux integral, rather than the true Riemann integral. Moreover, the definition is readily extended to defining Riemann–Stieltjes integration....
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  • Thumbnail for Cauchy's integral formula
    constant. We can use a combination of a Möbius transformation and the Stieltjes inversion formula to construct the holomorphic function from the real...
    25 KB (4,364 words) - 04:10, 17 May 2025
  • Laplace transform (category Integral transforms)
    }{2}}.} The (unilateral) Laplace–Stieltjes transform of a function g : ℝ → ℝ is defined by the Lebesgue–Stieltjes integral { L ∗ g } ( s ) = ∫ 0 ∞ e − s...
    76 KB (9,635 words) - 14:49, 12 July 2025
  • Thumbnail for Dirac delta function
    against a continuous function can be properly understood as a Riemann–Stieltjes integral: ∫ − ∞ ∞ f ( x ) δ ( d x ) = ∫ − ∞ ∞ f ( x ) d H ( x ) . {\displaystyle...
    97 KB (14,360 words) - 10:41, 13 July 2025
  • FX of X, with the expected value of g(X) now given by the Lebesgue–Stieltjes integral E ⁡ [ g ( X ) ] = ∫ − ∞ ∞ g ( x ) d F X ( x ) . {\displaystyle \operatorname...
    14 KB (2,099 words) - 20:40, 26 December 2024
  • Thumbnail for Partition of an interval
    Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. Specifically, as finer partitions of a given...
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  • of bounded variation. The Laplace–Stieltjes transform of F {\displaystyle F} is defined by the Stieltjes integral ω ( s ) = ∫ 0 ∞ e − s t d F ( t ) ...
    8 KB (1,413 words) - 21:11, 6 July 2025