Elementary Real and Complex Analysis

by Shilov, Georgi E.

• ISBN13: 9780486689227
• ISBN10: 0486689220
• Edition: Revised
• Format: Paperback
• Copyright: 1996-02-07
• Publisher: Dover Publications

Summary

Excellent undergraduate-level text offers coverage of real numbers, sets, metric spaces, limits, continuous functions, series, the derivative, higher derivatives, the integral and more. Each chapter contains a problem set (hints and answers at the end), while a wealth of examples and applications are found throughout the text. Over 340 theorems fully proved. 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