Book:Tom M. Apostol/Mathematical Analysis/Second Edition

Tom M. Apostol: Mathematical Analysis: A Modern Approach to Advanced Calculus

Published $\text {1973}$

Subject Matter

Analysis

Contents

Chapter 1 The Real and Complex Number Systems

Chapter 2 Some Basic Notions of Set Theory

Chapter 3 Elements of Point Set Topology

Chapter 4 Limits and Continuity

Chapter 5 Derivatives

Chapter 6 Functions of Bounded Variation and Rectifiable Curves

6.1 Introduction

6.2 Properties of monotonic functions

6.3 Functions of bounded variation

6.4 Total variation

6.5 Additive property of total variation

6.6 Total variation on $\closedint a x$ as a function of $x$

6.7 Functions of bounded variation expressed as the difference of increasing functions

6.8 Continuous functions of bounded variation

6.9 Curves and paths

6.10 Rectifiable paths and arc length

6.11 Additive and continuity properties of arc length

6.12 Equivalence of paths. Change of parameter

Exercises

Chapter 7 The Riemann-Stieltjes Integral

Chapter 8 Infinite Series and Infinite Products

Chapter 9 Sequences of Functions

9.1 Pointwise convergence of sequences of functions

9.2 Examples of sequences of real-valued functions

9.3 Definition of uniform convergence

9.4 Uniform convergence and continuity

9.5 The Cauchy condition for uniform convergence

9.6 Uniform convergence of infinite series of functions

9.7 A space-filling curve

9.8 Uniform convergence and Riemann-Stieltjes integration

9.9 Nonuniformly convergent sequences that can be integrated term by term

9.10 Uniform convergence and differentiation

9.11 Sufficient conditions for uniform convergence of a series

9.12 Uniform convergence and double sequences

9.13 Mean convergence

9.14 Power series

9.15 Multiplication of power series

9.16 The substitution theorem

9.17 Reciprocal of a power series

9.18 Real power series

9.19 The Taylor's series generated by a function

9.20 Bernstein's theorem

9.21 The binomial series

9.22 Abel's limit theorem

9.23 Tauber's theorem

Exercises

Chapter 10 The Lebesgue Integral

Chapter 11 Fourier Series and Fourier Integrals

Chapter 12 Multivariable Differential Calculus

Chapter 13 Implicit Functions and Extremum Problems

Chapter 14 Multiple Riemann Integrals

Chapter 15 Multiple Lebesgue Integrals

Chapter 16 Cauchy's Theorem and the Residue Calculus

Index of Special Symbols

Index

Further Editions

• 1957: Tom M. Apostol: Mathematical Analysis