A Modern View of the Riemann Integral

This monograph explores the full capabilities of the Riemann integral,based on three basic observations: a Riemann integrability criterion,admissible families of partitions,and modified Riemann sums. It is intended for enthusiasts with proof-based familiarity with the Riemann integral and mathematical maturity.

Format: Paperback / softback
Length: 176 pages
Publication date: 06 October 2022
Publisher: Springer International Publishing AG

This monograph delves into the full range of the Riemann integral, setting aside all notions from Lebesgue theory. Instead, it embarks on an exploration rooted in Riemann's original viewpoint. Along the way, readers will encounter numerous historical vignettes and discover a particular handiness for computations and applications.

The foundation of this approach lies in three fundamental observations. Firstly, 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. Secondly, 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 designed for enthusiasts eager to explore the potential of Riemann's original notion of integral. While a proof-based familiarity with the Riemann integral is the only formal prerequisite, readers will also need to draw upon mathematical maturity and a scholarly outlook.

This monograph offers a comprehensive exploration of the Riemann integral, shedding light on its rich capabilities and historical significance. By delving into Riemann's original viewpoint and employing novel approaches, readers will gain a deeper understanding of this fundamental mathematical concept and its applications in various fields. Whether you are a mathematician, engineer, or simply interested in the beauty of mathematics, this book will provide you with a valuable and enlightening journey into the world of the Riemann integral.

Weight: 296g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031117985
Edition number: 1st ed. 2022