Advertisement

© 2015

Understanding Analysis

Benefits

  • Provides a polished and tuned-up version of the same core text that has proved successful with students and instructors for 15 years

  • Includes around 150 new exercises, in addition to around 200 of the best exercises from the first edition, and an accompanying solutions manual for instructors

  • Presents three new self-guided projects exploring Euler’s sum, the factorial function and the Weierstrass Approximation Theorem

Textbook

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

  1. 1.Department of MathematicsMiddlebury CollegeMiddleburyUSA

About the authors

Stephen D. Abbott is Professor of Mathematics at Middlebury College. He is a two-time winner of Middlebury’s Perkins Award for Excellence in Teaching (1998, 2010). His published work includes articles in the areas of operator theory and functional analysis, the algorithmic foundations of robotics, and the intersection of science, mathematics and the humanities.          

Bibliographic information

  • Book Title Understanding Analysis
  • Authors Stephen Abbott
  • Series Title Undergraduate Texts in Mathematics
  • 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 Mathematics and Statistics (R0)
  • Hardcover ISBN 978-1-4939-2711-1
  • Softcover ISBN 978-1-4939-5026-3
  • eBook ISBN 978-1-4939-2712-8
  • Series ISSN 0172-6056
  • Series E-ISSN 2197-5604
  • Edition Number 2
  • Number of Pages XII, 312
  • Number of Illustrations 0 b/w illustrations, 36 illustrations in colour
  • Topics Analysis
  • Buy this book on publisher's site
Industry Sectors
Aerospace
Finance, Business & Banking
IT & Software

Reviews

“The choice of topics is a happy combination of the essential and the interesting, all truly leading to an understanding of what analysis is and what questions it addresses, aided by the author’s extraordinarily lucid exposition. … Summing Up: Highly recommended. Upper-division undergraduates.” (D. Robbins, Choice, Vol. 53 (2), October, 2015)

“This is the second edition of a text for an undergraduate course in single-variable real analysis. … The topics covered in this book are the ones that have, by now, become standard for a one-semester undergraduate real analysis course … . Overall, this book represents, to my mind, the gold standard among single-variable undergraduate analysis texts.” (Mark Hunacek, MAA Reviews, June, 2015)

“This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so well-motivated that 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; take a look at it next time you’re preparing that class.”

— Steve Kennedy, MAA Reviews

“Each chapter begins with a discussion section and ends with an epilogue. The discussion serves to motivate the content of the chapter while the epilogue points tantalisingly to more advanced topics. … I wish I had written this book! The development of the subject follows the tried-and-true path, but the presentation is engaging and challenging. Abbott focuses attention immediately on the topics which make analysis fascinating … and makes them accessible to an inexperienced audience.”

— Scott Sciffer, The Australian Mathematical Society Gazette