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A modern view of the Riemann integral

Torchinsky, Alberto.

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A modern view of the Riemann integral

Record Type:	Electronic resources : Monograph/item
Title/Author:	A modern view of the Riemann integral/ by Alberto Torchinsky.
Author:	Torchinsky, Alberto.
Published:	Cham :Springer International Publishing : : 2022.,
Description:	x, 176 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
Subject:	Riemann integral. -
Online resource:	https://doi.org/10.1007/978-3-031-11799-2
ISBN:	9783031117992

A modern view of the Riemann integral
Torchinsky, Alberto.

A modern view of the Riemann integral[electronic resource] /by Alberto Torchinsky. - Cham :Springer International Publishing :2022. - x, 176 p. :ill., digital ;24 cm. - Lecture notes in mathematics,v. 23091617-9692 ;. - Lecture notes in mathematics ;v. 2309..

This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue's theory, the author embarks on an exploration rooted in Riemann's original viewpoint. On this journey, we encounter new results, numerous historical vignettes, and discover a particular handiness for computations and applications. This approach rests on three basic observations. First, a Riemann integrability criterion in terms of oscillations, which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second, the introduction of the concepts of admissible families of partitions and modified Riemann sums. Finally, the fact that most numerical quadrature rules make use of carefully chosen Riemann sums, which makes the Riemann integral, be it proper or improper, most appropriate for this endeavor. A Modern View of the Riemann Integral is intended for enthusiasts keen to explore the potential of Riemann's original notion of integral. The only formal prerequisite is a proof-based familiarity with the Riemann integral, though readers will also need to draw upon mathematical maturity and a scholarly outlook.

ISBN: 9783031117992

Standard No.: 10.1007/978-3-031-11799-2doiSubjects--Topical Terms:

753036
Riemann integral.

LC Class. No.: QA311 / .T67 2022

Dewey Class. No.: 515.43

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245 1 2 $a A modern view of the Riemann integral $h [electronic resource] / $c by Alberto Torchinsky.
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490 1 $a Lecture notes in mathematics, $x 1617-9692 ; $v v. 2309
520 $a This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue's theory, the author embarks on an exploration rooted in Riemann's original viewpoint. On this journey, we encounter new results, numerous historical vignettes, and discover a particular handiness for computations and applications. This approach rests on three basic observations. First, a Riemann integrability criterion in terms of oscillations, which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second, the introduction of the concepts of admissible families of partitions and modified Riemann sums. Finally, the fact that most numerical quadrature rules make use of carefully chosen Riemann sums, which makes the Riemann integral, be it proper or improper, most appropriate for this endeavor. A Modern View of the Riemann Integral is intended for enthusiasts keen to explore the potential of Riemann's original notion of integral. The only formal prerequisite is a proof-based familiarity with the Riemann integral, though readers will also need to draw upon mathematical maturity and a scholarly outlook.
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