Softcover ISBN:  9781470456689 
Product Code:  AMSTEXT/47 
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eBook ISBN:  9781470460174 
Product Code:  AMSTEXT/47.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470456689 
eBook: ISBN:  9781470460174 
Product Code:  AMSTEXT/47.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 
Softcover ISBN:  9781470456689 
Product Code:  AMSTEXT/47 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470460174 
Product Code:  AMSTEXT/47.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470456689 
eBook ISBN:  9781470460174 
Product Code:  AMSTEXT/47.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 

Book DetailsPure and Applied Undergraduate TextsVolume: 47; 2020; 247 ppMSC: Primary 26
This is a text for students who have had a threecourse 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 selfcontained 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.
ReadershipUndergraduates interested in analysis in one variable.

Table of Contents

Cover

Title page

Copyright

Contents

Preface

Some basic notation

Chapter 1. Numbers

1.1. Peano arithmetic

1.2. The integers

1.3. Prime factorization and the fundamental theorem of arithmetic

1.4. The rational numbers

1.5. Sequences

1.6. The real numbers

1.7. Irrational numbers

1.8. Cardinal numbers

1.9. Metric properties of RR

1.10. Complex numbers

Chapter 2. Spaces

2.1. Euclidean spaces

2.2. Metric spaces

2.3. Compactness

2.4. The Baire category theorem

Chapter 3. Functions

3.1. Continuous functions

3.2. Sequences and series of functions

3.3. Power series

3.4. Spaces of functions

3.5. Absolutely convergent series

Chapter 4. Calculus

4.1. The derivative

4.2. The integral

4.3. Power series

4.4. Curves and arc length

4.5. The exponential and trigonometric functions

4.6. Unbounded integrable functions

Chapter 5. Further topics in analysis

5.1. Convolutions and bump functions

5.2. The Weierstrass approximation theorem

5.3. The StoneWeierstrass theorem

5.4. Fourier series

5.5. Newton’s method

5.6. Inner product spaces

Appendix A. Complementary results

A.1. The fundamental theorem of algebra

A.2. More on the power series of (1𝑥)^{𝑏}

A.3. 𝜋² is irrational

A.4. Archimedes’ approximation to 𝜋

A.5. Computing 𝜋 using arctangents

A.6. Power series for tan𝑥

A.7. Abel’s power series theorem

A.8. Continuous but nowheredifferentiable functions

Bibliography

Index

Back Cover


Additional Material

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This is a text for students who have had a threecourse 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 selfcontained 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.
Undergraduates interested in analysis in one variable.

Cover

Title page

Copyright

Contents

Preface

Some basic notation

Chapter 1. Numbers

1.1. Peano arithmetic

1.2. The integers

1.3. Prime factorization and the fundamental theorem of arithmetic

1.4. The rational numbers

1.5. Sequences

1.6. The real numbers

1.7. Irrational numbers

1.8. Cardinal numbers

1.9. Metric properties of RR

1.10. Complex numbers

Chapter 2. Spaces

2.1. Euclidean spaces

2.2. Metric spaces

2.3. Compactness

2.4. The Baire category theorem

Chapter 3. Functions

3.1. Continuous functions

3.2. Sequences and series of functions

3.3. Power series

3.4. Spaces of functions

3.5. Absolutely convergent series

Chapter 4. Calculus

4.1. The derivative

4.2. The integral

4.3. Power series

4.4. Curves and arc length

4.5. The exponential and trigonometric functions

4.6. Unbounded integrable functions

Chapter 5. Further topics in analysis

5.1. Convolutions and bump functions

5.2. The Weierstrass approximation theorem

5.3. The StoneWeierstrass theorem

5.4. Fourier series

5.5. Newton’s method

5.6. Inner product spaces

Appendix A. Complementary results

A.1. The fundamental theorem of algebra

A.2. More on the power series of (1𝑥)^{𝑏}

A.3. 𝜋² is irrational

A.4. Archimedes’ approximation to 𝜋

A.5. Computing 𝜋 using arctangents

A.6. Power series for tan𝑥

A.7. Abel’s power series theorem

A.8. Continuous but nowheredifferentiable functions

Bibliography

Index

Back Cover