A Radical Approach to Real Analysis

✍ Scribed by David Bressoud

• Publisher: American Mathematical Society
• Year: 2022
• Tongue: English
• Leaves: 453
• Series: AMS/MAA TEXTBOOKS 10
• Edition: 2
• Category: Library

✦ Synopsis

In this second edition of the MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on Infinite Summations, Differentiability and Continuity, and Convergence of Infinite Series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created. The book begins with Fourier's introduction of trigonometric series and the problems they created for the mathematicians of the early 19th century. It follows Cauchy's attempts to establish a firm foundation for calculus and considers his failures as well as his successes. It culminates with Dirichlet's proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet's proof.

✦ Table of Contents

Preface
Contents
1. Crisis in Mathematics: Fourier’s Series
1.1 Background to the Problem
1.2 Difficulties with the Solution
2. Infinite Summations
2.1 The Archimedean Understanding
2.2 Geometric Series.
2.3 Calculating π
2.4 Logarithms and the Harmonic Series
2.5 Taylor Series
2.6 Emerging Doubts
3. Differentiability and Continuity
3.1 Differentiability
3.2 Cauchy and the Mean Value Theorems
3.3 Continuity
3.4 Consequences of Continuity
3.5 Consequences of the Mean Value Theorem
4. The Convergence of Infinite Series
4.1 The Basic Tests of Convergence
4.2 Comparison Tests
4.3 The Convergence of Power Series
4.4 The Convergence of Fourier Series
5. Understanding Infinite Series
5.1 Groupings and Rearrangements
5.2 Cauchy and Continuity
5.3 Differentiation and Integration
5.4 Verifying Uniform Convergence
6. Return to Fourier Series
6.1 Dirichlet’s Theorem
6.2 The Cauchy Integral
6.3 The Riemann Integral
6.4 Continuity without Differentiability
7. Epilogue
A. Explorations of the Infinite
A.1 Wallis on π
A.2 Bernoulli’s Numbers
A.3 Sums of Negative Powers
A.4 The Size of n!
B. Bibliography
C. Hints to Selected Exercises
2.1.6
2.4.10
3.1.4
3.3.18
3.4.13
4.1.3
4.3.2
4.4.12
5.4.1
6.1.6
6.3.16
A.1.12
Index
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Corrections
Resources for
A Radical Approach to Real Analysis (2nd edition)
Chapter 1: Crises in Mathematics: Fourier's Series
Derivation of Fourier’s Solution
Laplace’s Equation
How Fourier found the coeffcients for equation (1.7)
Approximating Fourier's Solution (Maple code)
The General Solution
The Orthogonality Relation
Fourier Series as Complex Power Series
Maple code for exercises in section 1.2
Chapter 2: Infinite Summations
The Quadrature of the Parabolic Segment
The Archimedean Principle
Explorations of the Alternating Harmonic Series (Maple code)
Assigning Values to Divergent Series
More Pi (Maple code)
Newton’s Formula
Explorations of the Harmonic Series
Euler’s Solution to the Vibrating Drumhead
Explorations of d'Alembert's Series (Maple code)
Explorations of Lagrange's Remainder (Maple code)
Maple code for exercises in section 2.1
Maple code for exercises in section 2.2
Maple code for exercises in section 2.3
Maple code for exercises in section 2.4
Maple code for exercises in section 2.5
Chapter 3: Differentiability and Continuity
Newton-Raphson Method
How to find and write a proof
Continued Fractions
The Marquis de l’Hospital
Maple code for exercise in section 3.3
Maple code for exercises in section 3.4
Maple code for exercises in Newton-Raphson Method
Chapter 4: The Convergence of Infinite Series
Stirling's Formula (Maple code)
Exponential Function
Exponential Function (Maple code)
Convergence in Norm
Gauss’s Test
Maple code for exercises in section 4.1
Maple code for exercises in section 4.2
Maple code for exercises in section 4.3
Maple code for exercises in section 4.4
Chapter 5: Understanding Infinite Series
The Dilogarithm
Maple code for exercises in section 5.1
Maple code for exercises in section 5.2
Maple code for exercises in section 5.3
Chapter 6: Return to Fourier Series
Maple code for exercises in section 6.1
Maple code for exercises in section 6.2
Maple code for exercises in section 6.3
Maple code for exercises in section 6.4
Appendix A: Explorations of the Infinite
Binomial Coefficients and Sums of nth Powers
Maple code for exercises in section A.1
Maple code for exercises in section A.2
Maple code for exercises in section A.3
Maple code for exercises in section A.4
Acknowledgements
Back Cover