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Author: Ioannis Roussos
Publisher: CRC Press
ISBN: 146658808X
Category : Mathematics
Languages : en
Pages : 681
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Book Description
Improper Riemann Integrals is the first book to collect classical and modern material on the subject for undergraduate students. The book gives students the prerequisites and tools to understand the convergence, principal value, and evaluation of the improper/generalized Riemann integral. It also illustrates applications to science and engineering
Author: Ioannis Roussos
Publisher: CRC Press
ISBN: 146658808X
Category : Mathematics
Languages : en
Pages : 681
Get Book Here
Book Description
Improper Riemann Integrals is the first book to collect classical and modern material on the subject for undergraduate students. The book gives students the prerequisites and tools to understand the convergence, principal value, and evaluation of the improper/generalized Riemann integral. It also illustrates applications to science and engineering
Author: Robert M. McLeod
Publisher: American Mathematical Soc.
ISBN: 1614440204
Category : Riemann integral
Languages : en
Pages : 275
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Book Description
The Generalized Riemann Integral is addressed to persons who already have an acquaintance with integrals they wish to extend and to the teachers of generations of students to come. The organization of the work will make it possible for the first group to extract the principal results without struggling through technical details which they may find formidable or extraneous to their purposes. The technical level starts low at the opening of each chapter. Thus, readers may follow each chapter as far as they wish and then skip to the beginning of the next. To readers who do wish to see all the details of the arguments, they are given. The generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, haven't gotten a glimpse of such results as monotone and dominated convergence theorems. As its name hints, the generalized Riemann integral is defined in terms of Riemann sums. The path from the definition to theorems exhibiting the full power of the integral is direct and short.
Author: Open University. M431 Course Team
Publisher:
ISBN: 9780749220686
Category : Integrals, Generalized
Languages : en
Pages : 27
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Book Description
Author: Sheldon Axler
Publisher: Springer Nature
ISBN: 3030331431
Category : Mathematics
Languages : en
Pages : 430
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Book Description
This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online. For errata and updates, visit https://measure.axler.net/
Author: Alberto Torchinsky
Publisher: Springer Nature
ISBN: 3031117999
Category : Mathematics
Languages : en
Pages : 182
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Book Description
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.
Author: Douglas S. Kurtz
Publisher: World Scientific
ISBN: 9789812388438
Category : Mathematics
Languages : en
Pages : 286
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Book Description
This book presents a historical development of the integration theories of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, showing how new theories of integration were developed to solve problems that earlier theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and can be used separately in teaching a portion of an introductory course on real analysis. There is a sufficient supply of exercises to make the book useful as a textbook.
Author: Frank Burk
Publisher: MAA
ISBN: 9780883853375
Category : Mathematics
Languages : en
Pages : 312
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Book Description
The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there is a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reasons for their existence and their uses are given, with plentiful historical information. The audience for the book is advanced undergraduate mathematics students, graduate students, and faculty members, of which even the most experienced are unlikely to be aware of all of the integrals in the Garden of Integrals. Professor Burk's clear and well-motivated exposition makes this book a joy to read. There is no other book like it.
Author: Andrew Browder
Publisher: Springer Science & Business Media
ISBN: 1461207150
Category : Mathematics
Languages : en
Pages : 348
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Book Description
Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. To this extent, more than one proof is included for a theorem - where this is considered beneficial - so as to stimulate the students' reasoning for alternate approaches and ideas. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. Also included are some beautiful applications of this theory, such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions. Throughout, reference is made to earlier sections, so as to reinforce the main ideas by repetition. Unique in its applications to some topics not usually covered at this level.
Author: Ioannis Markos Roussos
Publisher: CRC Press
ISBN: 1466588071
Category : Mathematics
Languages : en
Pages : 692
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Book Description
Improper Riemann Integrals is the first book to collect classical and modern material on the subject for undergraduate students. The book gives students the prerequisites and tools to understand the convergence, principal value, and evaluation of the improper/generalized Riemann integral. It also illustrates applications to science and engineering problems. The book contains the necessary background, theorems, and tools, along with two lists of the most important integrals and sums computed in the text. Numerous examples at various levels of difficulty illustrate the concepts and theorems. The book uses powerful tools of real and complex analysis not only to compute the examples and solve the problems but also to justify that the computation methods are legitimate. Enriched with many examples, applications, and problems, this book helps students acquire a deeper understanding of the subject, preparing them for further study. It shows how to solve the integrals without exclusively relying on tables and computer packages.
Author: Robert G. Bartle
Publisher: American Mathematical Soc.
ISBN: 9780821883853
Category :
Languages : en
Pages : 480
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Book Description
The theory of integration is one of the twin pillars on which analysis is built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is ``better'' because it removes some restrictions on the integrands and the domains over which we integrate. However, there are still drawbacks to Lebesgue integration, for instance, dealing with the Fundamental Theorem of Calculus, or with ``improper'' integrals. This book is an introduction to a relatively new theory of the integral (called the ``generalized Riemann integral'' or the ``Henstock-Kurzweil integral'') that corrects the defects in the classical Riemann theory and both simplifies and extends the Lebesgue theory of integration. Although this integral includes that of Lebesgue, its definition is very close to the Riemann integral that is familiar to students from calculus. One virtue of the new approach is that no measure theory and virtually no topology is required. Indeed, the book includes a study of measure theory as an application of the integral. Part 1 fully develops the theory of the integral of functions defined on a compact interval. This restriction on the domain is not necessary, but it is the case of most interest and does not exhibit some of the technical problems that can impede the reader's understanding. Part 2 shows how this theory extends to functions defined on the whole real line. The theory of Lebesgue measure from the integral is then developed, and the author makes a connection with some of the traditional approaches to the Lebesgue integral. Thus, readers are given full exposure to the main classical results. The text is suitable for a first-year graduate course, although much of it can be readily mastered by advanced undergraduate students. Included are many examples and a very rich collection of exercises. There are partial solutions to approximately one-third of the exercises. A complete solutions manual is available separately.
