LECTURE NOTES ON RIEMANN INTEGRAL

LECTURE NOTES ON RIEMANN INTEGRAL
Title LECTURE NOTES ON RIEMANN INTEGRAL PDF eBook
Author కే వి వి విద్యా సాగర్
Publisher Principal, GDC Narsipatnam
Pages 12
Release
Genre
ISBN

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Lecture Notes on Riemann Integration

Lecture Notes on Riemann Integration
Title Lecture Notes on Riemann Integration PDF eBook
Author K V Vidyasagar
Publisher K V Vidyasagar
Pages 11
Release 2013-07-22
Genre
ISBN

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Title: Riemann Integration: Exploring Fundamental Principles Author: KUPARALA VENKATA VIDYASAGAR Dive into the world of Riemann integration with this comprehensive guide. This book offers a detailed exploration of the fundamental concepts, techniques, and applications of Riemann integration in the realm of mathematical analysis. From its inception by Bernhard Riemann to its modern interpretations and implications in various branches of mathematics and beyond, this text provides a clear and concise elucidation of this crucial mathematical tool. Inside these pages, readers will find: A rigorous yet accessible presentation of the Riemann integral, covering its definition, properties, and theorems. Practical examples and illustrative explanations aiding in the understanding of Riemann integration and its applications in calculus and beyond. Discussions on the convergence of Riemann sums, the Riemann integrability of functions, and connections to other areas of mathematics, including differential equations and complex analysis. Insightful exercises and problems to reinforce understanding and encourage further exploration. Whether you're a student delving into real analysis, a mathematician seeking a deeper comprehension of integration principles, or an enthusiast curious about the foundations of calculus, this book serves as an invaluable resource, offering a comprehensive and insightful journey into the world of Riemann integration.

A Modern View of the Riemann Integral

A Modern View of the Riemann Integral
Title A Modern View of the Riemann Integral PDF eBook
Author Alberto Torchinsky
Publisher Springer Nature
Pages 182
Release 2022-10-05
Genre Mathematics
ISBN 3031117999

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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.

The Lebesgue Integral

The Lebesgue Integral
Title The Lebesgue Integral PDF eBook
Author Open University. M431 Course Team
Publisher
Pages 27
Release 1992
Genre Integrals, Generalized
ISBN 9780749220686

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Measure and Integration

Measure and Integration
Title Measure and Integration PDF eBook
Author M Thamban Nair
Publisher CRC Press
Pages 194
Release 2019-11-06
Genre Mathematics
ISBN 1000739872

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This concise text is intended as an introductory course in measure and integration. It covers essentials of the subject, providing ample motivation for new concepts and theorems in the form of discussion and remarks, and with many worked-out examples. The novelty of Measure and Integration: A First Course is in its style of exposition of the standard material in a student-friendly manner. New concepts are introduced progressively from less abstract to more abstract so that the subject is felt on solid footing. The book starts with a review of Riemann integration as a motivation for the necessity of introducing the concepts of measure and integration in a general setting. Then the text slowly evolves from the concept of an outer measure of subsets of the set of real line to the concept of Lebesgue measurable sets and Lebesgue measure, and then to the concept of a measure, measurable function, and integration in a more general setting. Again, integration is first introduced with non-negative functions, and then progressively with real and complex-valued functions. A chapter on Fourier transform is introduced only to make the reader realize the importance of the subject to another area of analysis that is essential for the study of advanced courses on partial differential equations. Key Features Numerous examples are worked out in detail. Lebesgue measurability is introduced only after convincing the reader of its necessity. Integrals of a non-negative measurable function is defined after motivating its existence as limits of integrals of simple measurable functions. Several inquisitive questions and important conclusions are displayed prominently. A good number of problems with liberal hints is provided at the end of each chapter. The book is so designed that it can be used as a text for a one-semester course during the first year of a master's program in mathematics or at the senior undergraduate level. About the Author M. Thamban Nair is a professor of mathematics at the Indian Institute of Technology Madras, Chennai, India. He was a post-doctoral fellow at the University of Grenoble, France through a French government scholarship, and also held visiting positions at Australian National University, Canberra, University of Kaiserslautern, Germany, University of St-Etienne, France, and Sun Yat-sen University, Guangzhou, China. The broad area of Prof. Nair’s research is in functional analysis and operator equations, more specifically, in the operator theoretic aspects of inverse and ill-posed problems. Prof. Nair has published more than 70 research papers in nationally and internationally reputed journals in the areas of spectral approximations, operator equations, and inverse and ill-posed problems. He is also the author of three books: Functional Analysis: A First Course (PHI-Learning, New Delhi), Linear Operator Equations: Approximation and Regularization (World Scientific, Singapore), and Calculus of One Variable (Ane Books Pvt. Ltd, New Delhi), and he is also co-author of Linear Algebra (Springer, New York).

Lecture Notes in Real Analysis

Lecture Notes in Real Analysis
Title Lecture Notes in Real Analysis PDF eBook
Author Xiaochang Wang
Publisher Springer
Pages 217
Release 2018-11-21
Genre Mathematics
ISBN 3319989561

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This compact textbook is a collection of the author’s lecture notes for a two-semester graduate-level real analysis course. While the material covered is standard, the author’s approach is unique in that it combines elements from both Royden’s and Folland’s classic texts to provide a more concise and intuitive presentation. Illustrations, examples, and exercises are included that present Lebesgue integrals, measure theory, and topological spaces in an original and more accessible way, making difficult concepts easier for students to understand. This text can be used as a supplementary resource or for individual study.

Measure Theory and Integration

Measure Theory and Integration
Title Measure Theory and Integration PDF eBook
Author G De Barra
Publisher Elsevier
Pages 240
Release 2003-07-01
Genre Mathematics
ISBN 0857099523

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This text approaches integration via measure theory as opposed to measure theory via integration, an approach which makes it easier to grasp the subject. Apart from its central importance to pure mathematics, the material is also relevant to applied mathematics and probability, with proof of the mathematics set out clearly and in considerable detail. Numerous worked examples necessary for teaching and learning at undergraduate level constitute a strong feature of the book, and after studying statements of results of the theorems, students should be able to attempt the 300 problem exercises which test comprehension and for which detailed solutions are provided. - Approaches integration via measure theory, as opposed to measure theory via integration, making it easier to understand the subject - Includes numerous worked examples necessary for teaching and learning at undergraduate level - Detailed solutions are provided for the 300 problem exercises which test comprehension of the theorems provided