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Classical Analysis: An Approach through Problems

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A conceptually clear induction to fundamental analysis theorems, a tutorial for creative approaches for solving problems, a collection of modern challenging problems, a pathway to undergraduate research―all these desires gave life to the pages here.

This book exposes students to stimulating and enlightening proofs and hard problems of classical analysis mainly published in The American Mathematical Monthly.

The author presents proofs as a form of exploration rather than just a manipulation of symbols. Drawing on the papers from the Mathematical Association of America's journals, numerous conceptually clear proofs are offered. Each proof provides either a novel presentation of a familiar theorem or a lively discussion of a single issue, sometimes with multiple derivations.

The book collects and presents problems to promote creative techniques for problem-solving and undergraduate research and offers instructors an opportunity to assign these problems as projects. This book provides a wealth of opportunities for these projects.

Each problem is selected for its natural charm―the connection with an authentic mathematical experience, its origination from the ingenious work of professionals, develops well-shaped results of broader interest.

Author(s): Hongwei Chen
Series: Textbooks in Mathematics
Publisher: CRC Press/Chapman & Hall
Year: 2022

Language: English
Pages: 442
City: Boca Raton

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
1. Sequences
1.1. Completeness Theorems for the Real Number System
1.2. Stolz-Cesàro Theorem
1.3. Worked Examples
1.3.1. ε – N definition
1.3.2. Cauchy criterion
1.3.3. The squeeze theorem
1.3.4. Monotone convergence theorem
1.3.5. Upper and lower limits
1.3.6. Stolz-Cesàro theorem
1.3.7. Fixed-point theorems
1.3.8. Recursions with closed forms
1.3.9. Limits involving the harmonic numbers
1.4. Exercises
2. Infinite Numerical Series
2.1. Main Definitions and Basic Convergence Tests
2.2. Raabe and Logarithmic Tests
2.3. The Kummer, Bertrand, and Gauss Tests
2.4. More Sophisticated Tests Based on Monotonicity
2.5. On the Universal Test
2.6. Tests for General Series
2.7. Properties of Convergent Series
2.8. Infinite Products
2.9. Worked Examples
2.10. Exercises
3. Continuity
3.1. Definition of Continuity
3.2. Limits of Functions
3.3. Three Fundamental Theorems
3.4. From the Intermediate Value Theorem to Chaos
3.5. Monotone Functions
3.6. Worked Examples
3.7. Exercises
4. Differentiation
4.1. Derivatives
4.2. Fundamental Theorems of Differentiation
4.3. L'Hôpital's Rules
4.4. Convex Functions
4.5. Taylor's Theorem
4.6. Worked Examples
4.7. Exercises
5. Integration
5.1. The Riemann Integral
5.2. Classes of Integrable Functions
5.3. The Mean Value Theorem
5.4. The Fundamental Theorems of Calculus
5.5. Worked Examples
5.6. Exercises
6. Sequences and Series of Functions
6.1. Pointwise and Uniform Convergence
6.2. Importance of Uniform Convergence
6.3. Two Other Convergence Theorems
6.4. Power Series
6.5. Weierstrass's Approximation Theorem
6.6. Worked Examples
6.7. Exercises
7. Improper and Parametric Integration
7.1. Improper Integrals
7.2. Integrals with Parameters
7.3. The Gamma Function
7.4. Worked Examples
7.5. Exercises
A. List of Problems from MAA
Bibliography
Index