Synopses & Reviews
In this book the renowned Russian mathematician Georgi E. Shilov brings his unique perspective to real and complex analysis, an area of perennial interest in mathematics. Although there are many books available on the topic, the present work is specially designed for undergraduates in mathematics, science and engineering. A high level of mathematical sophistication is not required.
The book begins with a systematic study of real numbers, understood to be a set of objects satisfying certain definite axioms. The concepts of a mathematical structure and an isomorphism are introduced in Chapter 2, after a brief digression on set theory, and a proof of the uniqueness of the structure of real numbers is given as an illustration. Two other structures are then introduced, namely n-dimensional space and the field of complex numbers.
After a detailed treatment of metric spaces in Chapter 3, a general theory of limits is developed in Chapter 4. Chapter 5 treats some theorems on continuous numerical functions on the real line, and then considers the use of functional equations to introduce the logarithm and the trigonometric functions. Chapter 6 is on infinite series, dealing not only with numerical series but also with series whose terms are vectors and functions (including power series). Chapters 7 and 8 treat differential calculus proper, with Taylor's series leading to a natural extension of real analysis into the complex domain. Chapter 9 presents the general theory of Riemann integration, together with a number of its applications. Analytic functions are covered in Chapter 10, while Chapter 11 is devoted to improper integrals, and makes full use of the technique of analytic functions.
Each chapter includes a set of problems, with selected hints and answers at the end of the book. A wealth of examples and applications can be found throughout the text. Over 340 theorems are fully proved.
Synopsis
Excellent undergraduate-level text offers coverage of real numbers, sets, metric spaces, limits, continuous functions, much more. Each chapter contains a problem set with hints and answers. 1973 edition.
Table of Contents
Preface
1 Real Numbers
1.1. Set-Theoretic Preliminaries
1.2. Axioms for the Real Number System
1.3. Consequences of the Addition Axioms
1.4. Consequences of the Multiplication Axioms
1.5. Consequences of the Order Axioms
1.6. Consequences of the Least Upper Bound Axiom
1.7. The Principle of Archimedes and Its Consequences
1.8. The Principle of Nested Intervals
1.9. The Extended Real Number System
Problems
2 Sets
2.1. Operations on Sets
2.2. Equivalence of Sets
2.3. Countable Sets
2.4 Uncountable Sets
2.5. Mathematical Structures
2.6. n-Dimensional Space
2.7. Complex Numbers
2.8. Functions and Graphs
Problems
3 Metric Spaces
3.1. Definitions and Examples
3.2. Open Sets
3.3. Convergent Sequences and Homeomorphisms
3.4. Limit Points
3.5. Closed Sets
3.6. Dense Sets and Closures
3.7. Complete Metric Spaces
3.8. Completion of a Metric Space
3.9. Compactness
Problems
4 Limits
4.1. Basic Concepts
4.2. Some General Theorems
4.3. Limits of Numerical Functions
4.4. Upper and Lower Limits
4.5. Nondecreasing and Nonincreasing Functions
4.6. Limits of Numerical Functions
4.7. Limits of Vector Functions
Problems
5 Continuous Functions
5.1. Continuous Functions on a Metric Space
5.2. Continuous Numerical Functions on the Real Line
5.3. Monotonic Functions
5.4. The Logarithm
5.5. The Exponential
5.6. Trignometric Functions
5.7. Applications of Trigonometric Functions
5.8. Continuous Vector Functions of a Vecor Variable
5.9. Sequences of Functions
Problems
6 Series
6.1. Numerical Series
6.2. Absolute and Conditional Convergences
6.3. Operations on Series
6.4. Series of Vectors
6.5. Series of Functions
6.6. Power Series
Problems
7 The Derivative
7.1. Definitions and Examples
7.2. Properties of Differentiable Functions
7.3. The Differential
7.4. Mean Value Theorems
7.5. Concavity and Inflection Points
7.6. L'Hospital's Rules
Problems
8 Higher Derivatives
8.1. Definitions and Examples
8.2. Taylor's Formula
8.3. More on Concavity and Inflection Points
8.4. Another Version of Taylor's Formula
8.5. Taylor Series
8.6. Complex Exponentials and Trigonometric Functions
8.7. Hyperbolic Functions
Problems
9 The Integral
9.1. Definitions and Basic Properties
9.2. Area and Arc Length
9.3. Antiderivatives and Indefinite Integrals
9.4. Technique of Indefinite Integrals
9.5. Evaluation of Definite Integrals
9.6. More on Area
9.7. More on Arc Length
9.8. Area of a Surface of Revolution
9.9. Further Applications of Integration
9.10. Integration of Sequences of Functions
9.11. Parameter-Dependent Integrals
9.12. Line Integrals
Problems
10 Analytic Functions
10.1. Basic Concepts
10.2. Line Integrals of Complex Functions
10.3. Cauchy's Theorem and Its Consequences
10.4. Residues and Isolated Singular Points
10.5. Mappings and Elementary Functions
Problems
11 Improper Integrals
11.1. Improper Integralsof the First Kind
11.2. Convergence of Improper Integrals
11.3. Improper Integrals of the Second and Third Kinds
11.4 Evaluation of Improper Integrals by Residues
11.5 Parameter-Dependent ImproperIntegrals
11.6 The Gamma and Beta Functions
Problems
Appendix A Elementary Symbolic Logic
Appendix B Measure and Integration on a Compact Metric Space
Selected Hints and Answers
Index