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Understanding Analysis

  • Stephen Abbott

Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Stephen Abbott
    Pages 1-37
  3. Stephen Abbott
    Pages 39-84
  4. Stephen Abbott
    Pages 85-109
  5. Stephen Abbott
    Pages 111-144
  6. Stephen Abbott
    Pages 145-167
  7. Stephen Abbott
    Pages 169-213
  8. Stephen Abbott
    Pages 215-248
  9. Stephen Abbott
    Pages 249-303
  10. Back Matter
    Pages 305-312

About this book

Introduction

This lively introductory text exposes the student to the rewards of a rigorous study of functions of a real variable. In each chapter, informal discussions of questions that give analysis its inherent fascination are followed by precise, but not overly formal, developments of the techniques needed to make sense of them. By focusing on the unifying themes of approximation and the resolution of paradoxes that arise in the transition from the finite to the infinite, the text turns what could be a daunting cascade of definitions and theorems into a coherent and engaging progression of ideas. Acutely aware of the need for rigor, the student is much better prepared to understand what constitutes a proper mathematical proof and how to write one.

Fifteen years of classroom experience with the first edition of Understanding Analysis have solidified and refined the central narrative of the second edition. Roughly 150 new exercises join a selection of the best exercises from the first edition, and three more project-style sections have been added. Investigations of Euler’s computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book’s cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis.

Review of the first edition:

“This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated t

hat exposing students to it could well lead them to expect such excellence in all their textbooks. … Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen. … This terrific book will become the text of choice for the single-variable introductory analysis course … ”

— Steve Kennedy, MAA Reviews           

Keywords

Abbott analysis Baire Category Theorem Calculus Henstock integral Riemann integral Taylor series Weierstrass approximation theorem derivatives fundamental theorem of Calculus gamma function general topology generalized Riemann integral intermediate value theorem mean value theorem power series real analysis real numbers

Authors and affiliations

  • Stephen Abbott
    • 1
  1. 1.Department of MathematicsMiddlebury CollegeMiddleburyUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4939-2712-8
  • Copyright Information Springer Science+Business Media New York 2015
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4939-2711-1
  • Online ISBN 978-1-4939-2712-8
  • Series Print ISSN 0172-6056
  • Series Online ISSN 2197-5604
  • Buy this book on publisher's site