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Problems in Mathematical Analysis III: Integration
W. J. Kaczor and M. T. Nowak, Marie Curie-Sklodowska University, Lublin, Poland
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Student Mathematical Library
2003; 356 pp; softcover
Volume: 21
ISBN-10: 0-8218-3298-0
ISBN-13: 978-0-8218-3298-1
List Price: US$54
Institutional Members: US$43.20
All Individuals: US$43.20
Order Code: STML/21
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The best way to penetrate the subtleties of the theory of integration is by solving problems. This book, like its two predecessors, is a wonderful source of interesting and challenging problems. As a resource, it is unequaled. It offers a much richer selection than is found in any current textbook. Moreover, the book includes a complete set of solutions.

This is the third volume of Problems in Mathematical Analysis. The topic here is integration for real functions of one real variable. The first chapter is devoted to the Riemann and the Riemann-Stieltjes integrals. Chapter 2 deals with Lebesgue measure and integration.

The authors include some famous, and some not so famous, inequalities related to Riemann integration. Many of the problems for Lebesgue integration concern convergence theorems and the interchange of limits and integrals. The book closes with a section on Fourier series, with a concentration on Fourier coefficients of functions from particular classes and on basic theorems for convergence of Fourier series.

The book is mainly geared toward students studying the basic principles of analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of the material is designed to help student comprehension, to encourage them to ask their own questions, and to start research. The collection of problems will also help teachers who wish to incorporate problems into their lectures. The problems are grouped into sections according to the methods of solution. Solutions for the problems are provided.

Problems in Mathematical Analysis I and II are available as Volumes 4 and 12 in the AMS series, Student Mathematical Library.

Readership

Undergraduates, graduate students, and instructors interested in analysis.

Reviews

From reviews for Volumes I and II:

"A valuable resource."

-- American Mathematical Monthly

"Would be an ideal choice for tutorial or problem-solving seminars. The volume is also suitable for self-study ... presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research ... a really useful book for practice in mathematical analysis."

-- Zentralblatt MATH

"Belongs to the great tradition of Eastern European problem books ... if you love mathematics and are serious about understanding analysis, this book is a must."

-- MAA Online

"A very stimulating problem book ... The style ... is proven to be a motivating approach in constructing and conveying mathematical knowledge ... leads the readers to find new solutions and hence boosts their ability to carry out further research ... thorough coverage of some topics that are covered very briefly in other compatible books ... of interest to anyone who wishes to pursue research in mathematical analysis and its applications ... also excellent for students who want to enhance their skills in real analysis ... a useful supplement to any graduate textbook in mathematical analysis ... some problems are also suitable for undergraduate students."

-- MAA Online

Table of Contents

Problems
  • The Riemann-Stieltjes integral
  • The Lebesgue integral
Solutions
  • The Riemann-Stieltjes integral
  • The Lebesgue integral
  • Bibliography
  • Index
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