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Introduction to Analysis in One Variable
 
Michael E. Taylor :  University of North Carolina, Chapel Hill, Chapel Hill, NC
Introduction to Analysis in One Variable
Softcover ISBN:  978-1-4704-5668-9
Product Code:  AMSTEXT/47
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6017-4
Product Code:  AMSTEXT/47.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-5668-9
eBook: ISBN:  978-1-4704-6017-4
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
Add to cart
Introduction to Analysis in One Variable
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Introduction to Analysis in One Variable
Michael E. Taylor :  University of North Carolina, Chapel Hill, Chapel Hill, NC
Softcover ISBN:  978-1-4704-5668-9
Product Code:  AMSTEXT/47
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
eBook ISBN:  978-1-4704-6017-4
Product Code:  AMSTEXT/47.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-5668-9
eBook ISBN:  978-1-4704-6017-4
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
Add to cart
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 47; 2020; 247 pp
    MSC: Primary 26

    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
     
     
    • 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 Stone-Weierstrass 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 (1-𝑥)^{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 nowhere-differentiable functions
    • Bibliography
    • Index
    • Back Cover
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Pure and Applied Undergraduate Texts
Volume: 47; 2020; 247 pp
MSC: Primary 26

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.
  • 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 Stone-Weierstrass 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 (1-𝑥)^{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 nowhere-differentiable functions
  • Bibliography
  • Index
  • Back Cover
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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