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Improper Riemann Integrals
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Book Synopsis Improper Riemann Integrals by : Ioannis Markos Roussos
Download or read book Improper Riemann Integrals written by Ioannis Markos Roussos and published by CRC Press. This book was released on 2016-04-19 with total page 689 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Improper Riemann Integrals by : Ioannis Markos Roussos
Download or read book Improper Riemann Integrals written by Ioannis Markos Roussos and published by CRC Press. This book was released on 2013-12-16 with total page 692 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Improper Riemann Integrals by : Ioannis Roussos
Download or read book Improper Riemann Integrals written by Ioannis Roussos and published by CRC Press. This book was released on 2023-09-27 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: The scope of this book is the improper or generalized Riemann integral and infinite sum (series). The reader will study its convergence, principal value, evaluation and application to science and engineering. Improper Riemann integrals and infinite sums are interconnected. In the new edition, the author has involved infinite sums more than he did in the first edition. Apart from having computed and listed a large number of improper integrals and infinite sums, we have also developed the necessary theory and various ways of evaluating them or proving their divergence. Questions, problems and applications involving various improper integrals and infinite sums (series) of numbers emerge in science and application very often. Their complete presentations and all rigorous proofs would require taking the graduate-level courses on these subjects. Here their statements are adjusted to a level students of all levels can understand and use them efficiently as powerful tools in a large list of problems and applications.
Book Synopsis The Lebesgue Integral by : Open University. M431 Course Team
Download or read book The Lebesgue Integral written by Open University. M431 Course Team and published by . This book was released on 1992 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Generalized Riemann Integral by : Robert M. McLeod
Download or read book The Generalized Riemann Integral written by Robert M. McLeod and published by American Mathematical Soc.. This book was released on 1980-12-31 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis A Modern Theory of Integration by : Robert G. Bartle
Download or read book A Modern Theory of Integration written by Robert G. Bartle and published by American Mathematical Soc.. This book was released on 2001-03-21 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Active Calculus 2018 by : Matthew Boelkins
Download or read book Active Calculus 2018 written by Matthew Boelkins and published by Createspace Independent Publishing Platform. This book was released on 2018-08-13 with total page 560 pages. Available in PDF, EPUB and Kindle. Book excerpt: Active Calculus - single variable is a free, open-source calculus text that is designed to support an active learning approach in the standard first two semesters of calculus, including approximately 200 activities and 500 exercises. In the HTML version, more than 250 of the exercises are available as interactive WeBWorK exercises; students will love that the online version even looks great on a smart phone. Each section of Active Calculus has at least 4 in-class activities to engage students in active learning. Normally, each section has a brief introduction together with a preview activity, followed by a mix of exposition and several more activities. Each section concludes with a short summary and exercises; the non-WeBWorK exercises are typically involved and challenging. More information on the goals and structure of the text can be found in the preface.
Book Synopsis Measure Theory by : Vladimir I. Bogachev
Download or read book Measure Theory written by Vladimir I. Bogachev and published by Springer Science & Business Media. This book was released on 2007-01-15 with total page 1075 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.
Book Synopsis The Theory of Lebesgue Measure and Integration by : S. Hartman
Download or read book The Theory of Lebesgue Measure and Integration written by S. Hartman and published by Elsevier. This book was released on 2014-07-14 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Theory of Lebesgue Measure and Integration deals with the theory of Lebesgue measure and integration and introduces the reader to the theory of real functions. The subject matter comprises concepts and theorems that are now considered classical, including the Yegorov, Vitali, and Fubini theorems. The Lebesgue measure of linear sets is discussed, along with measurable functions and the definite Lebesgue integral. Comprised of 13 chapters, this volume begins with an overview of basic concepts such as set theory, the denumerability and non-denumerability of sets, and open sets and closed sets on the real line. The discussion then turns to the theory of Lebesgue measure of linear sets based on the method of M. Riesz, together with the fundamental properties of measurable functions. The Lebesgue integral is considered for both bounded functions — upper and lower integrals — and unbounded functions. Later chapters cover such topics as the Yegorov, Vitali, and Fubini theorems; convergence in measure and equi-integrability; integration and differentiation; and absolutely continuous functions. Multiple integrals and the Stieltjes integral are also examined. This book will be of interest to mathematicians and students taking pure and applied mathematics.
Book Synopsis Calculus of One Variable by : M. Thamban Nair
Download or read book Calculus of One Variable written by M. Thamban Nair and published by Springer Nature. This book was released on 2022-01-22 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is designed to serve as a textbook for courses offered to undergraduate and graduate students enrolled in Mathematics. The first edition of this book was published in 2015. As there is a demand for the next edition, it is quite natural to take note of the several suggestions received from the users of the earlier edition over the past six years. This is the prime motivation for bringing out a revised second edition with a thorough revision of all the chapters. The book provides a clear understanding of the basic concepts of differential and integral calculus starting with the concepts of sequences and series of numbers, and also introduces slightly advanced topics such as sequences and series of functions, power series, and Fourier series which would be of use for other courses in mathematics for science and engineering programs. The salient features of the book are - precise definitions of basic concepts; several examples for understanding the concepts and for illustrating the results; includes proofs of theorems; exercises within the text; a large number of problems at the end of each chapter as home-assignments. The student-friendly approach of the exposition of the book would be of great use not only for students but also for the instructors. The detailed coverage and pedagogical tools make this an ideal textbook for students and researchers enrolled in a mathematics course.
Book Synopsis Undergraduate Analysis by : Serge Lang
Download or read book Undergraduate Analysis written by Serge Lang and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 651 pages. Available in PDF, EPUB and Kindle. Book excerpt: This logically self-contained introduction to analysis centers around those properties that have to do with uniform convergence and uniform limits in the context of differentiation and integration. From the reviews: "This material can be gone over quickly by the really well-prepared reader, for it is one of the book’s pedagogical strengths that the pattern of development later recapitulates this material as it deepens and generalizes it." --AMERICAN MATHEMATICAL SOCIETY
Book Synopsis Measure and Integration Theory by : Heinz Bauer
Download or read book Measure and Integration Theory written by Heinz Bauer and published by Walter de Gruyter. This book was released on 2011-04-20 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem. The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory. The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.
Book Synopsis Theories of Integration by : Douglas S. Kurtz
Download or read book Theories of Integration written by Douglas S. Kurtz and published by World Scientific. This book was released on 2004 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Mathematical Analysis Fundamentals by : Agamirza Bashirov
Download or read book Mathematical Analysis Fundamentals written by Agamirza Bashirov and published by Academic Press. This book was released on 2014-03-27 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author’s goal is a rigorous presentation of the fundamentals of analysis, starting from elementary level and moving to the advanced coursework. The curriculum of all mathematics (pure or applied) and physics programs include a compulsory course in mathematical analysis. This book will serve as can serve a main textbook of such (one semester) courses. The book can also serve as additional reading for such courses as real analysis, functional analysis, harmonic analysis etc. For non-math major students requiring math beyond calculus, this is a more friendly approach than many math-centric options. Friendly and well-rounded presentation of pre-analysis topics such as sets, proof techniques and systems of numbers. Deeper discussion of the basic concept of convergence for the system of real numbers, pointing out its specific features, and for metric spaces Presentation of Riemann integration and its place in the whole integration theory for single variable, including the Kurzweil-Henstock integration Elements of multiplicative calculus aiming to demonstrate the non-absoluteness of Newtonian calculus.
Book Synopsis Aspects of Integration by : Ronald B. Guenther
Download or read book Aspects of Integration written by Ronald B. Guenther and published by CRC Press. This book was released on 2023-08-24 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aspects of Integration: Novel Approaches to the Riemann and Lebesgue Integrals is comprised of two parts. The first part is devoted to the Riemann integral, and provides not only a novel approach, but also includes several neat examples that are rarely found in other treatments of Riemann integration. Historical remarks trace the development of integration from the method of exhaustion of Eudoxus and Archimedes, used to evaluate areas related to circles and parabolas, to Riemann’s careful definition of the definite integral, which is a powerful expansion of the method of exhaustion and makes it clear what a definite integral really is. The second part follows the approach of Riesz and Nagy in which the Lebesgue integral is developed without the need for any measure theory. Our approach is novel in part because it uses integrals of continuous functions rather than integrals of step functions as its starting point. This is natural because Riemann integrals of continuous functions occur much more frequently than do integrals of step functions as a precursor to Lebesgue integration. In addition, the approach used here is natural because step functions play no role in the novel development of the Riemann integral in the first part of the book. Our presentation of the Riesz-Nagy approach is significantly more accessible, especially in its discussion of the two key lemmas upon which the approach critically depends, and is more concise than other treatments. Features Presents novel approaches designed to be more accessible than classical presentations A welcome alternative approach to the Riemann integral in undergraduate analysis courses Makes the Lebesgue integral accessible to upper division undergraduate students How completion of the Riemann integral leads to the Lebesgue integral Contains a number of historical insights Gives added perspective to researchers and postgraduates interested in the Riemann and Lebesgue integrals
Book Synopsis Fundamental Mathematical Analysis by : Robert Magnus
Download or read book Fundamental Mathematical Analysis written by Robert Magnus and published by Springer Nature. This book was released on 2020-07-14 with total page 445 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook offers a comprehensive undergraduate course in real analysis in one variable. Taking the view that analysis can only be properly appreciated as a rigorous theory, the book recognises the difficulties that students experience when encountering this theory for the first time, carefully addressing them throughout. Historically, it was the precise description of real numbers and the correct definition of limit that placed analysis on a solid foundation. The book therefore begins with these crucial ideas and the fundamental notion of sequence. Infinite series are then introduced, followed by the key concept of continuity. These lay the groundwork for differential and integral calculus, which are carefully covered in the following chapters. Pointers for further study are included throughout the book, and for the more adventurous there is a selection of "nuggets", exciting topics not commonly discussed at this level. Examples of nuggets include Newton's method, the irrationality of π, Bernoulli numbers, and the Gamma function. Based on decades of teaching experience, this book is written with the undergraduate student in mind. A large number of exercises, many with hints, provide the practice necessary for learning, while the included "nuggets" provide opportunities to deepen understanding and broaden horizons.
Book Synopsis Analysis in Euclidean Space by : Kenneth Hoffman
Download or read book Analysis in Euclidean Space written by Kenneth Hoffman and published by Courier Dover Publications. This book was released on 2019-07-17 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: Developed for an introductory course in mathematical analysis at MIT, this text focuses on concepts, principles, and methods. Its introductions to real and complex analysis are closely formulated, and they constitute a natural introduction to complex function theory. Starting with an overview of the real number system, the text presents results for subsets and functions related to Euclidean space of n dimensions. It offers a rigorous review of the fundamentals of calculus, emphasizing power series expansions and introducing the theory of complex-analytic functions. Subsequent chapters cover sequences of functions, normed linear spaces, and the Lebesgue interval. They discuss most of the basic properties of integral and measure, including a brief look at orthogonal expansions. A chapter on differentiable mappings addresses implicit and inverse function theorems and the change of variable theorem. Exercises appear throughout the book, and extensive supplementary material includes a Bibliography, List of Symbols, Index, and an Appendix with background in elementary set theory.
