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

Download or read book Theories Of Integration The Integrals Of Riemann Lebesgue Henstock kurzweil And Mcshane written by Charles W Swartz and published by World Scientific Publishing Company. This book was released on 2004-06-03 with total page 283 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.

Download or read book Theories Of Integration The Integrals Of Riemann Lebesgue Henstock kurzweil And Mcshane 2nd Edition written by Charles W Swartz and published by World Scientific Publishing Company. This book was released on 2011-10-31 with total page 311 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book uses classical problems to motivate 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 integration 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 could be used separately in teaching a portion of an introductory real analysis course. There is a sufficient supply of exercises to make this book useful as a textbook.

Download or read book Lectures on the Theory of Integration written by Ralph Henstock and published by World Scientific. This book was released on 1988 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended to be self-contained, giving the theory of absolute (equivalent to Lebesgue) and non-absolute (equivalent to Denjoy-Perron) integration by using a simple extension of the Riemann integral. A useful tool for mathematicians and scientists needing advanced integration theory would be a method combining the ideas of the calculus of indefinite integral and Riemann definite integral in such a way that Lebesgue properties can be proved easily.Three important results that have not appeared in any other book distinguish this book from the rest. First a result on limits of sequences under the integral sign, secondly the necessary and sufficient conditions for the various limits under the integral sign and thirdly the application of these results to ordinary differential equations. The present book will give non-absolute integration theory just as easily as the absolute theory, and Stieltjes-type integration too.

Download or read book Riemann s and Lebesgue s Integral written by Rubin Schatten and published by . This book was released on 1939 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A Riemann Type Integral that Includes Lebesgue Stieltjes Bochner and Stochastic Integrals written by Edward James McShane and published by American Mathematical Soc.. This book was released on 1969 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the difficulties with integration theory is that there are so many different detailed definitions that the non-expert is confused about their relative strengths and usefulness. A surprising recent development in the theory of integration has been the discovery that suitable modifications to the Riemann definition using approximating sums can produce a wide variety of different integrals including integrals of great power.

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 70 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Lebesgue Stieltjes Integral written by M. Carter and published by . This book was released on 2000-05-01 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Theories of Integration written by Douglas S. Kurtz and published by World Scientific Publishing Company Incorporated. This book was released on 2004-01-01 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: - Contains a thorough treatment of each of the integrals of Riemann, Lebesgue, Henstock-Kurzweil and McShane - Discusses the weaknesses of the various integrals and presents a comparison of the integrals - Abundant supply of exercises and examples - Chapters can be used independently

Download or read book The Generalized Riemann Integral written by Robert M. McLeod and published by MAA Press. This book was released on 1980 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The General Theory of Integration written by Ralph Henstock and published by . This book was released on 1991 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Every good mathematical book stands like a tree with its roots in the past and its branches stretching out towards the future. Whether the fruits of this tree are desirable and whether the branches will be quarried for mathematical wood to build further edifices, I will leave to the judgment of history. The roots of this book take nourishment from the concept of definite integration of continuous functions, where Riemann's method is the high water mark of the simpler theory.

Download or read book The Bochner Integral written by J. Mikusinski and published by Birkhäuser. This book was released on 2014-08-23 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of the Lebesgue integral is still considered as a difficult theory, no matter whether it is based the concept of measure or introduced by other methods. The primary aim of this book is to give an approach which would be as intelligible and lucid as possible. Our definition, produced in Chapter I, requires for its background only a little of the theory of absolutely convergent series so that it is understandable for students of the first undergraduate course. Nevertheless, it yields the Lebesgue integral in its full generality and, moreover, extends automatically to the Bochner integral (by replacing real coefficients of series by elements of a Banach space). It seems that our approach is simple enough as to eliminate the less useful Riemann integration theory from regular mathematics courses. Intuitively, the difference between various approaches to integration may be brought out by the following story on shoemakers. A piece of leather, like in Figure 1, is given. The task consists in measuring its area. There are three shoemakers and each of them solves the task in his own way. A B Fig. 1 The shoemaker R. divides the leather into a finite number of vertical strips and considers the strips approximately as rectangles. The sum of areas of all rectangles is taken for an approximate area of the leather (Figure 2). If he is not satisfied with the obtained exactitude, he repeats the whole procedure, by dividing the leather into thinner strips.

Download or read book Concerning the Riemann and Lebesgue Integrals written by Mabel J. Foster and published by . This book was released on 1940 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Generalized Riemann Integral written by Robert M. MacLeod and published by . This book was released on 1980 with total page 275 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Recursive Functions written by Rózsa Péter and published by . This book was released on 1967 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Large Sieve written by Klaus Friedrich Roth and published by . This book was released on 1969 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Radiosity and Realistic Image Synthesis written by Michael F. Cohen and published by Elsevier. This book was released on 2012-12-02 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of image synthesis is to create, using the computer, a visual experience that is identical to what a viewer would experience when viewing a real environment. Radiosity and Realistic Image Synthesis offers the first comprehensive look at the radiosity method for image synthesis and the tools required to approach this elusive goal. Basic concepts and mathematical fundamentals underlying image synthesis and radiosity algorithms are covered thoroughly. (A basic knowledge of undergraduate calculus is assumed). The algorithms that have been developed to implement the radiosity method ranging from environment subdivision to final display are discussed. Successes and difficulties in implementing and using these algorithms are highlighted. Extensions to the basic radiosity method to include glossy surfaces, fog or smoke, and realistic light sources are also described. There are 16 pages of full colour images and over 100 illustrations to explain the development and show the results of the radiosity method. Results of applications of this new technology from a variety of fields are also included. Michael Cohen has worked in the area of realistic image synthesis since 1983 and was instrumental in the development of the radiosity method. He is currently an assistant professor of computer science at Princeton University. John Wallace is a software engineer at 3D/EYE, Inc., where he is the project leader for the development of Hewlett-Packard's ATRCore radiosity and ray tracing library. A chapter on the basic concepts of image synthesis is contributed by Patrick Hanrahan. He has worked on the topic of image synthesis at Pixar, where he was instrumental in the development of the Renderman software. He has also led research on the hierarchical methods at Princeton University, where he is an associate professor of computer science. All three authors have written numerous articles on radiosity that have appeared in the SIGGAPH proceedings and elsewhere. They have also taught the SIGGRAPH course on radiosity for 5 years. * The first comprehensive book written about radiosity - Features applications from the fields of computer graphics, architecture, industrial design, and related computer aided design technologies - Offers over 100 illustrations and 16 pages of full-color images demonstrating the results of radiosity methods - Contains a chapter authored by Pat Hanrahan on the basic concepts of image synthesis and a foreword by Donald Greenberg