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Principles of Mathematical Analysis

by: Rudin, Walter

Principles of Mathematical Analysis

by: Rudin, Walter

ISBN 13:
9780070542358
ISBN 10:
007054235X
Edition: 3rd
Format: Hardcover
Copyright: 01/01/1976
Publisher: McGraw Hill

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Summary

The third edition of this well known text continues to provide a solid foundation in mathematical analysis for undergraduate and first-year graduate students. The text begins with a discussion of the real number system as a complete ordered field. (Dedekind's construction is now treated in an appendix to Chapter I.) The topological background needed for the development of convergence, continuity, differentiation and integration is provided in Chapter 2. There is a new section on the gamma function, and many new and interesting exercises are included.This text is part of the Walter Rudin Student Series in Advanced Mathematics.

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Principles of Mathematical Analysis

Principles of Mathematical Analysis

9780070542358

007054235X

Summary

Table of Contents

Introduction

Ordered Sets

Fields

The Real Field

The Extended Real Number System

The Complex Field

Euclidean Spaces

Appendix

Exercises

Chapter 2: Basic Topology

Finite, Countable, and Uncountable Sets

Metric Spaces

Compact Sets

Perfect Sets

Connected Sets

Exercises

Chapter 3: Numerical Sequences and Series

Convergent Sequences

Subsequences

Cauchy Sequences

Upper and Lower Limits

Some Special Sequences

Series

Series of Nonnegative Terms

The Number e

The Root and Ratio Tests

Power Series

Summation by Parts

Absolute Convergence

Addition and Multiplication of Series

Rearrangements

Exercises

Chapter 4: Continuity

Limits of Functions

Continuous Functions

Continuity and Compactness

Continuity and Connectedness

Discontinuities

Monotonic Functions

Infinite Limits and Limits at Infinity

Exercises

Chapter 5: Differentiation

The Derivative of a Real Function

Mean Value Theorems

The Continuity of Derivatives

L'Hospital's Rule

Derivatives of Higher-Order

Taylor's Theorem

Differentiation of Vector-valued Functions

Exercises

Chapter 6: The Riemann-Stieltjes Integral

Definition and Existence of the Integral

Properties of the Integral

Integration and Differentiation

Integration of Vector-valued Functions

Rectifiable Curves

Exercises

Chapter 7: Sequences and Series of Functions

Discussion of Main Problem

Uniform Convergence

Uniform Convergence and Continuity

Uniform Convergence and Integration

Uniform Convergence and Differentiation

Equicontinuous Families of Functions

The Stone-Weierstrass Theorem

Exercises

Chapter 8: Some Special Functions

Power Series

The Exponential and Logarithmic Functions

The Trigonometric Functions

The Algebraic Completeness of the Complex Field

Fourier Series