mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition...

known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval...

framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be...

Montel's theorem Riemann–Stieltjes integral Stieltjes constants Stieltjes matrix Stieltjes moment problem Stieltjes polynomials Stieltjes transformation...

central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators...

Lebesgue–Stieltjes integral, further developed by Johann Radon, which generalizes both the Riemann–Stieltjes and Lebesgue integrals. The Daniell integral, which...

he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex...

Wikipedia page. Length Area Volume Probability Moving average Riemann sum Riemann–Stieltjes integral Bounded variation Jordan content Cauchy principal value...

The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform...

Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced [dɑ̃ʒwa]), Luzin integral or Perron...

Measure Sigma-algebra Lebesgue space Lebesgue–Stieltjes integration Riemann integral Henstock–Kurzweil integral This approach can be found in most treatments...

corresponding Lebesgue–Stieltjes measure, and the integral with respect to the Lebesgue–Stieltjes measure agrees with the Riemann–Stieltjes integral for continuous...

theorem Riemann–Stieltjes integral Riemann series theorem Riemann sum Riemann–von Mangoldt formula Riemann hypothesis Generalized Riemann hypothesis...

: 13–15  Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered. The first integral, with broad...

the style of a Riemann–Stieltjes integral). Many integration techniques of ordinary calculus can be used for the Stratonovich integral, e.g.: if f : R...

same, but using the Riemann–Stieltjes integral, along with an appropriate function of bounded variation, gives a definition of integral equivalent to the...

Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. Specifically, as finer partitions of...

Riemann–Stieltjes integration. Darboux integrals are named after their inventor, Gaston Darboux (1842–1917). The definition of the Darboux integral considers...

The formula is derived by applying integration by parts for a Riemann–Stieltjes integral to the functions A {\displaystyle A} and ϕ {\displaystyle \phi...

of the Cauchy–Riemann equations. Note that for smooth complex-valued functions f of compact support on C the generalized Cauchy integral formula simplifies...

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

using the Riemann–Stieltjes integral, and where F {\displaystyle F} is the cumulative distribution function. This is simply the Laplace-Stieltjes transform...

variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions. Another characterization states...

covers topics such as continuous functions, differentiation, the Riemann–Stieltjes integral, sequences and series of functions (in particular uniform convergence)...

{\displaystyle f(t)=\int _{0}^{\infty }e^{-tx}\,dg(x),} the integral being a Riemann–Stieltjes integral. In more abstract language, the theorem characterises...

{R} }\lambda \,\mathrm {d} E_{\lambda }\,.} The integral is understood as a Riemann–Stieltjes integral, convergent with respect to the norm on B(H). In...

the n-th moment of the probability distribution is given by the Riemann–Stieltjes integral μ n ′ = E ⁡ [ X n ] = ∫ − ∞ ∞ x n d F ( x ) {\displaystyle \mu...

against a continuous function can be properly understood as a Riemann–Stieltjes integral: ∫ − ∞ ∞ f ( x ) δ ( d x ) = ∫ − ∞ ∞ f ( x ) d H ( x ) . {\displaystyle...

bounded on the interval (0, A] for some 0 ≤ α < 1. The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many...

mathematics, the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function:...