Other integrals
Although the Riemann and Lebesgue integrals are the most important definitions of the integral, a number of others exist, including:
- The Riemann-Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes, an extension of the Riemann integral.
- The Lebesgue-Stieltjes integral In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework, further developed by Johann Radon Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on calculus of variations (in 1910, at the University of Vienna), which generalizes the Riemann-Stieltjes In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes and Lebesgue integrals In mathematics, Lebesgue integration refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a sub-domain of the real line or a higher dimensional Euclidean space with respect to the Lebesgue measure. This article focuses on the more general.
- The Daniell integral One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results for the integral can be obtained. However, an alternative approach is available, developed by Percy J. Daniell in his 1918 paper "A general form of integral&, which subsumes the Lebesgue integral In mathematics, Lebesgue integration refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a sub-domain of the real line or a higher dimensional Euclidean space with respect to the Lebesgue measure. This article focuses on the more general and Lebesgue-Stieltjes integral In measure-theoretic analysis and related branches of mathematics, Lebesgue-Stieltjes integration generalizes Riemann-Stieltjes and Lebesgue integration, preserving the many advantages of the latter in a more general measure-theoretic framework without the dependence on measures.
- The Henstock-Kurzweil integral In mathematics, the Henstock–Kurzweil integral, also known as the Denjoy integral and the Perron integral, is a possible definition of the integral of a function. It is a generalization of the Riemann integral which in some situations is more useful than the Lebesgue integral, variously defined by Arnaud Denjoy Arnaud Denjoy was a French mathematician, Oskar Perron Oskar Perron was a German mathematician, and (most elegantly, as the gauge integral) Jaroslav Kurzweil Jaroslav Kurzweil (Czech pronunciation: [ˈjarɔslaf ˈkurtsvajl]) is a Czech mathematician. He is a specialist in ordinary differential equations and defined the Perron integral in terms of Riemann sums. Kurzweil has been awarded the highest possible scientific prize of the Czech Republic, the "Czech Mind" of the year 2006, as an, and developed by Ralph Henstock Ralph Henstock was an English mathematician and author. As an Integration theorist, he is notable for Henstock-Kurzweil integral. Henstock brought the theory to a highly developed stage without ever having encountered Jaroslav Kurzweil's 1957 paper on the subject. Robert Bartle[5] gave perhaps the most compelling introduction to this integral in a paper for which he earned a writing award from the Mathematical Association of America.
- The Itō integral Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion . It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itō stochastic integral and Stratonovich integral In stochastic processes, the Stratonovich integral is a stochastic integral, the most common alternative to the Itō integral. While the Ito integral is the usual choice in applied math, the Stratonovich integral is frequently used in physics, which define integration with respect to stochastic processes such as Brownian motion In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called Brownian motion, after Robert Brown. It is one of the best known Lévy processes and occurs frequently in pure and applied mathematics, economics and physics.
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