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

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## Tom M. Apostol:

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

Published $\text {1974}$

### Subject Matter

### 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