Elementary Real and Complex Analysis

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