3rd Edition

# An Introduction to Analysis

By James R. Kirkwood Copyright 2022
330 Pages 55 B/W Illustrations
by Chapman & Hall

336 Pages 55 B/W Illustrations
by Chapman & Hall

Also available as eBook on:

The third edition of this widely popular textbook is authored by a master teacher. This book provides a mathematically rigorous introduction to analysis of real­valued functions of one variable. This intuitive, student-friendly text is written in a manner that will help to ease the transition from primarily computational to primarily theoretical mathematics.

The material is presented clearly and as intuitive as possible while maintaining mathematical integrity. The author supplies the ideas of the proof and leaves the write-up as an exercise. The text also states why a step in a proof is the reasonable thing to do and which techniques are recurrent.

Examples, while no substitute for a proof, are a valuable tool in helping to develop intuition and are an important feature of this text. Examples can also provide a vivid reminder that what one hopes might be true is not always true.

Features of the Third Edition:

• Begins with a discussion of the axioms of the real number system.

• The limit is introduced via sequences.
• Examples motivate what is to come, highlight the need for hypothesis in a theorem, and make abstract ideas more concrete.
• A new section on the Cantor set and the Cantor function.
• Exercises range in difficulty from the routine "getting your feet wet" types of problems to the moderately challenging problems.
• Topology of the real number system is developed to obtain the familiar properties of continuous functions.
• Some exercises are devoted to the construction of counterexamples.

The author presents the material to make the subject understandable and perhaps exciting to those who are beginning their study of abstract mathematics.

Preface

Introduction

1. The Real Number System
2. Sequences of Real Numbers
3. Topology of the Real Numbers
4. Continuous Functions
5. Differentiation
6. Integration
7. Series of Real Numbers
8. Sequences and Series of Functions
9. Fourier Series

Bibliography

Hints and Answers to Selected Exercises

Index

Biography

James R. Kirkwood holds a Ph.D. from University of Virginia. He has authored fifteen, published mathematics textbooks on various topics including calculus, real analysis, mathematical biology and mathematical physics. His original research was in mathematical physics, and he co-authored the seminal paper in a topic now called Kirkwood-Thomas Theory in mathematical physics. During the summer, he teaches real analysis to entering graduate students at the University of Virginia. He has been awarded several National Science Foundation grants. His texts, Elementary Linear Algebra, Linear Algebra, and Markov Processes, are also published by CRC Press.

Preface. Introduction. 1. The Real Number System. 1.1. Sets and Functions. 1.2. Properties of the Real Numbers as an Ordered Field. 1.3. The Completeness Axiom. 2. Sequences of Real Numbers. 2.1. Sequences of Real Numbers. 2.2. Subsequences. 2.3. The Bolzano-Weierstrass Theorem. 3. Topology of the Real Numbers. 3.1. Topology of the Real Numbers. 4. Continuous Functions. 4.1. Limits and continuity. 4.2. Monotone and Inverse Functions. 5. Differentiation. 5.1. The Derivative of a Function. 5.2. Some Mean Value Theorems. 6. Integration. 6.1. The Riemann Integral. 6.2. Some properties and applications of the Riemann Integral. 6.3. The Riemann-Stieltjes Integral. 7. Series of Real Numbers. 7.1. Tests for Convergence of Series. 7.2. Operations Involving Series. 8. Sequences and Series of Functions. 8.1. Sequences of functions. 8.2. Series of Functions. 8.3. Taylor Series. 8.4. The Cantor Set and Cantor Function. 9. Fourier Series. 9.1. Fourier Coefficients. 9.2. Representation by Fourier Series. Bibliography. Hints and Answers for Selected Exercises. Index.

### Biography

James R. Kirkwood holds a Ph.D. from University of Virginia. He has authored fifteen, published mathematics textbooks on various topics including calculus, real analysis, mathematical biology and mathematical physics. His original research was in mathematical physics, and he co-authored the seminal paper in a topic now called Kirkwood-Thomas Theory in mathematical physics. During the summer, he teaches real analysis to entering graduate students at the University of Virginia. He has been awarded several National Science Foundation grants. His texts, Elementary Linear Algebra, Linear Algebra, and Markov Processes, are also published by CRC Press.