**Pure and Applied Undergraduate Texts**

Volume: 47;
2020;
247 pp;
Softcover

MSC: Primary 26;

**Print ISBN: 978-1-4704-5668-9
Product Code: AMSTEXT/47**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

**Electronic ISBN: 978-1-4704-6017-4
Product Code: AMSTEXT/47.E**

List Price: $85.00

AMS Member Price: $68.00

MAA Member Price: $76.50

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#### Supplemental Materials

# Introduction to Analysis in One Variable

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*Michael E. Taylor*

This is a text for students who have had a three-course calculus sequence and who are ready to explore the logical structure of analysis as the backbone of calculus. It begins with a development of the real numbers, building this system from more basic objects (natural numbers, integers, rational numbers, Cauchy sequences), and it produces basic algebraic and metric properties of the real number line as propositions, rather than axioms. The text also makes use of the complex numbers and incorporates this into the development of differential and integral calculus. For example, it develops the theory of the exponential function for both real and complex arguments, and it makes a geometrical study of the curve \((\mathrm{exp}\thinspace it)\), for real \(t\), leading to a self-contained development of the trigonometric functions and to a derivation of the Euler identity that is very different from what one typically sees. Further topics include metric spaces, the Stone–Weierstrass theorem, and Fourier series.

#### Readership

Undergraduates interested in analysis in one variable.

#### Table of Contents

# Table of Contents

## Introduction to Analysis in One Variable

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents v7
- Preface vii9
- Some basic notation xi13
- Chapter 1. Numbers 115
- 1.1. Peano arithmetic 216
- 1.2. The integers 822
- 1.3. Prime factorization and the fundamental theorem of arithmetic 1327
- 1.4. The rational numbers 1529
- 1.5. Sequences 1933
- 1.6. The real numbers 2640
- 1.7. Irrational numbers 3650
- 1.8. Cardinal numbers 4054
- 1.9. Metric properties of RR 4559
- 1.10. Complex numbers 5064

- Chapter 2. Spaces 6175
- Chapter 3. Functions 8195
- Chapter 4. Calculus 107121
- Chapter 5. Further topics in analysis 177191
- Appendix A. Complementary results 219233
- A.1. The fundamental theorem of algebra 219233
- A.2. More on the power series of (1-𝑥)^{𝑏} 221235
- A.3. 𝜋² is irrational 222236
- A.4. Archimedes’ approximation to 𝜋 224238
- A.5. Computing 𝜋 using arctangents 228242
- A.6. Power series for tan𝑥 232246
- A.7. Abel’s power series theorem 234248
- A.8. Continuous but nowhere-differentiable functions 238252

- Bibliography 243257
- Index 245259
- Back Cover Back Cover1264