Understanding analysis

By: Abbott, Stephen
Material type: TextTextSeries: Undergraduate texts in mathematicsPublisher: New York Springer 2016Edition: 2ndDescription: xii, 312 p. Includes bibliographical reference and indexISBN: 9781493927111Subject(s): Mathematical analysis | Global analysis (Mathematics) | FunctionsDDC classification: 515 Summary: 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 rigour, 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. https://www.springer.com/gp/book/9781493927111
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Slot 1370 (0 Floor, East Wing) Non-fiction 515 A2U6 (Browse shelf) Available 201527

Table of Content

Preface
1 The Real Numbers
2 Sequences and Series
3 Basic Topology of R
4 Functional Limits and Continuity
5 The Derivative
6 Sequences and Series of Functions
7 The Riemann Integral
8 Additional Topics
Bibliography
Index.

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 rigour, 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.

https://www.springer.com/gp/book/9781493927111

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