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

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

Published $\text {1973}$.

### 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