Riemann-Stieltjes integral

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	Riemann integral - Wikipedia, the free encyclopedia
		An integral which is in fact a direct generalization of the Riemann integral is the Henstock-Kurzweil integral.
		The definition of the Lebesgue integral is not obviously a generalization of the Riemann integral, but it is not hard to prove that every Riemann-integrable function is Lebesgue-integrable and that the values of the two integrals agree whenever they are both defined.
		In the branch of mathematics 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.
	en.wikipedia.org /wiki/Riemann_integral (2271 words)

	Riemann-Stieltjes integral - Wikipedia, the free encyclopedia
		In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.
		With this definition, an integral can exist when f and g share points of discontinuity, as long as they are not discontinuous from the same side at the same point.
		and the existence of the integral on the left implies the existence of the integral on the right.
	en.wikipedia.org /wiki/Riemann-Stieltjes_integral (545 words)

	Inequalities for Riemann-Stieltjes Integrals
		J.V. Herod, A Gronwall inequality for linear Stieltjes integrals, Proc.
		Groh, A nonlinear Volterra-Stieltjes integral equation and a Gronwall inequality in one dimension, Illinois J. Math., 24(2) (1980), 244-263.
		A.M. Dcyachkov, On the existence of the Stieltjes integral, (Russian) Dokl.
	rgmia.vu.edu.au /ineq_riemann.htm (524 words)

	PlanetMath: Riemann-Stieltjes integral
		This is version 3 of Riemann-Stieltjes integral, born on 2002-07-23, modified 2002-07-27.
		exists, then it is unique and is known as the Riemann-Stieltjes integral of
		Cross-references: integral, finer, Riemann integrable, sum, partition, interval, finite, closed, functions, bounded
	planetmath.org /encyclopedia/RiemannStieltjesIntegral.html (79 words)

	7.1. Riemann Integral
		Show that the constant function f(x) = c is Riemann integrable on any interval [a, b] and find the value of the integral.
		Riemann sums have the practical disadvantage that we do not know which point to take inside each subinterval.
		The third example shows that not every function is Riemann integrable, and the second one shows that we need an easier condition to determine integrability of a given function.
	www.shu.edu /projects/reals/integ/riemann.html (1802 words)

	Moment-generating function - Wikipedia, the free encyclopedia
		Regardless of whether the probability distribution is continuous or not, the moment-generating function is given by the Riemann-Stieltjes integral
	en.wikipedia.org /wiki/Moment-generating_function (202 words)

	PlanetMath: contour integral
		Cross-references: piecewise smooth, continuous, rectifiable, integral, sign of, right, variable, complex, components, terms, Riemann-Stieltjes integral, between, Riemann integral, real axis, segment, infinity, limit, sum, partition, curve, image, function
		(i) If is a segment of the real axis, then this definition reduces to that of the Riemann integral of f(x) between
		This is version 8 of contour integral, born on 2002-07-24, modified 2005-03-18.
	planetmath.org /encyclopedia/ContourIntegral.html (238 words)

	Riemann Sums
		Riemann sums for integrals of the moduli of certain trigonometric polynomials.
		Riemann sums for integrals of moduli of certain trigonometric polynomials.
		Riemann sums for stochastic integrals and Lp moduli of continuity.
	mathews.ecs.fullerton.edu /n2003/riemannsum/RiemannSumBib/Links/RiemannSumBib_lnk_3.html (1002 words)

	Existence of Riemann-Stieltjes Integrals
		R., A criterion for the convergence of the Stieltjes integral.
		Va\u\i nerman, Sufficient conditions for the existence of the Stieltjes integral, (Russian) Vestnik Leningrad.
		P.P. Korovkin, The Stieltjes integral, and change of variable, (Russian) Application of functional analysis in approximation theory, 1 (Russian), pp.
	rgmia.vu.edu.au /riemann_stieltjes.htm (229 words)

	Sever S.
		Results for univariate and multivariate real functions and their natural applications in the error analysis of numerical quadratures for both simple and multiple integrals as well as for the Riemann-Stieltjes integral are given.
		Integral inequalities involving functions with bounded derivatives, otherwise known as Ostrowski-type integral inequalities, have enjoyed a surge in popularity.
		This book is intended for researchers and graduate students working in the fields of integral inequalities, approximation theory, applied mathematics, probability theory and stochastics, and numerical analysis.
	www.yurinsha.com /347/p12.htm (581 words)

	Springer Online Reference Works
		The Stieltjes integral has the property of linearity relative to both the integrand and the integrating function (given the condition that every one of the Stieltjes integrals on the right-hand side exists):
		is of bounded variation, the value of the Stieltjes integral coincides with the value of the corresponding Lebesgue–Stieltjes integral.
		Th.J. Stieltjes [1] hit upon the idea of such an integral when studying the positive "distribution of masses" on a straight line defined by an increasing function
	eom.springer.de /s/s087800.htm (365 words)

	RS_int
		I was doing the Riemann integral and not the Stieltjes integral.
		Thus, I used integral f dg as a shorthand for integral f g' dx.
		But you are also correct that in order to write integral f(x) d g(x) = integral f(x) g'(x) dx, we need g to be absolutely continuous.
	www.math.niu.edu /~rusin/known-math/00_incoming/RS_int (313 words)

	Integration by Parts: a Visual Proof
		Under that assumption, Riemann-Stieltjes integral reduces to Riemann integral, and
		, this definition is compatible with the definition of the Riemann integral (when
		If is a Riemann integrable function with respect to
	www.mi.sanu.ac.yu /vismath/sasho (383 words)

	A new necessary condition for existence of the Stieltjes integral, V. Chernyatin
		The paper is devoted to establishing a necessary condition for the existence of the Riemann-Stieltjes integral of continuous functions and simultaneously to finding a limit form of its relationship with the classical Riemann integral depending on a parameter.
		A new necessary condition for existence of the Stieltjes integral
		The mathematical technique for this study is based on changing the weighted first difference of a given function into the simple difference of a function determined by a first-order difference equation.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.rae/1122482128 (182 words)

	Riemann-Stieljes
		the stieljes integral is the one where alpha(x) is any increasing function; the riemann integral is the special case where alpha(x)=x.
		For I am told it is Riemann who in a paper, introduced the know definition of the Riemann-Stieljes' integral in terms of partitions, upper and lower integral, and probably also the Riemann sum caracterisation.
		The integral may or may not exist though, there are various theorems on existence that I can't usually remember, like f continuous and alpha of bounded variation will do it, but this isn't required.
	www.physicsforums.com /showthread.php?t=110841 (488 words)

	m151a
		Topics covered in the sequence include the real and complex number systems, sequences and series, continuity, differentiation, the Riemann-Stieltjes integral, sequences and series of functions, functions of several variables, and an introduction to Lebesgue integration.
		Definition of the integral, conditions guaranteeing existence, properties of the integral, integration and differentiation, vector valued functions, rectifiable curves.
		This is the first course in a three quarter sequence giving a rigorous development of mathematical analysis.
	math.ucr.edu /home/UndergradInfo/pages/m151a (189 words)

	Wilmott Forums - Riemann-Stieltjes Integral
		The 1st formula for Riemann-Stieltjes in my original post is then shown to be a method of calculating the Ito Integral, which seems a much better way of introducing ito calculus than other books (who simply show the two versions of Riemann limit, and say one is martigale and other is not).
		Strictly speaking, neither of the above integrals is a RS integral.
		This seems to be the key point which leads to the two Riemann sums not agreeing.
	www.wilmott.com /messageview.cfm?catid=8&threadid=33746 (909 words)

	Atlas: New Bounds for the Generalised Trapezoidal Rule for the Riemann-Stieltjes Integral and Applications by Pietro Cerone
		The solution to many problems in applied probability requires the evaluation of Riemann-Stieltjes integrals involving the convolution of cumulative distribution functions.Closed form expressions for the solution are very rare indeed.
		Atlas: New Bounds for the Generalised Trapezoidal Rule for the Riemann-Stieltjes Integral and Applications by Pietro Cerone
		The evaluation of Stieltjes integrals on which this paper concentrates is thus,it is argued, germane to the solution of such problems.
	atlas-conferences.com /c/a/g/d/98.htm (195 words)

	Citebase - New Inequalities of Gruss Type for the Stieltjes Integral and Applications
		[6] S.S. DRAGOMIR, On the Ostrowski inequality for Riemann-Stieltjes integral where f is of Holder type and u is of bounded variation and applications, J. DRAGOMIR, The Ostrowski's integral inequality for Lipschitzian mappings and applications, Computers and Mathematics with Applications, 38 (1999), 33-37.
		Sharp inequalitieis of Gruss type for Stieltjes integrals with application in numerical integration are provided.
		New Inequalities of Gruss Type for the Stieltjes Integral and Applications
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0303381 (634 words)

	anades
		For example, students should be familiar with the theory of the Riemann integral (but not necessarily the Riemann-Stieltjes integral), the Fundamental Theorem of Calculus, and the Lesbesgue criterion for Riemann integrability.
		Another good reference for the Riemann integral is `Calculus' by M. Spivak.
		The Riemann integral, including the Fundamental Theorem of Calculus and Lebesgue's criterion for Riemann integrability
	cams.njit.edu /Graduate/Qualifying_exams/analysis.html (344 words)

	About The Book
		Foundations of Mathematical Analysis covers real analysis—from naïve set theory and the axioms for the real numbers to the Lebesgue integral, with sequences and series, metric spaces, the Riemann-Stieltjes integral, inner product spaces, Fourier series, Tauberian theorems, the Riesz representation theorem, and a brief discussion of Hilbert spaces in between.
		We discuss measure zero (Section 57) and give a necessary and sufficient condition for the existence of the Riemann integral (Section 58).
		The last chapter (Chapter XIV) studies the Lebesgue integral.
	condor.depaul.edu /~rjohnson/foma/about.html (757 words)

	MATH 252 EXAM 2 INFORMATION
		The restricted linearity of the Riemann-Stieltjes integral in the ``dF''
		Definition of of the Darboux-Stieltjes Integral, the ``jump'' sum, the upper and lower sums.
		Be able to evaluate the Stieltjes integral for particular example.
	www.math.wvu.edu /~sherm/m252/exam2.html (421 words)

	RS_Existence.html
		This shows that the Riemann-Stieltjes integral for this pair does not exist.
		By the definition of the upper Riemann-Stieltjes integral as a infimum of upper sums, we can find a partition P
		to be discontinuous but for the Reiemann-Stieltjes integral of f to still exist.
	www.math.sc.edu /~sharpley/math555/Lectures/RS_Existence.html (365 words)

	Expectation
		We have defined the Riemann-Stieltjes integral in a context of expectation.
		, this reduces to the ordinary Riemann integral.
		has an integral in the latter sense (which is equal to 0), but not in the former sense.
	www.engineering.usu.edu /classes/ece/6010/lecture2/node1.html (452 words)

	MATHS 472: Real Analysis 2 (3)
		The Riemann-Stieltjes integral and the Fundamental Theorem of Calculus. Sequences of functions. Multiple Riemann integrals and change of variable.
		The Riemann-Stieltjes integral and Fundamental Theorem of Calculus. Sequences of functions. Multiple Riemann integrals.
		Students completing the sequence Maths 471-472 will gain a deep understanding of most aspects of calculus. They will understand more profoundly the connections among calculus, geometry, and linear algebra. They will gain experience in writing proofs and solving problems related to the content of the course.
	www.bsu.edu /math/ugsyllabi/472.htm (358 words)

	Newsgroops - Riemann-Stieltjes integral help
		I can prove it the other way (the other 2 integrals existing)
		ensure that both (a,b) and (b,c) of the integral exist just because the
		I need to prove that if S [c a] f dg exists and a
	www.newsgroops.org /group/sci.math/article-281861.html (111 words)

	Georgia Tech School of Mathematics: Courses Offered
		The Riemann Stieltjes integral, integration by parts, change of variables.
		Differentiation of functions of one real variable, Riemann-Stieltjes integral, the derivative in R
		: Mean value theorem, iterated integrals, and change of variables.
	www.math.gatech.edu /academic/courses?class=4318 (71 words)

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