# Elementary Real and Complex Analysis

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## Description

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.

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## Product details

- Paperback | 528 pages
- 137.16 x 213.36 x 27.94mm | 544.31g
- 07 Feb 1996
- Dover Publications Inc.
- New York, United States
- English
- Revised
- New ed of 2 Revised ed
- Illustrations, unspecified
- 0486689220
- 9780486689227
- 158,997

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## Table of contents

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

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