# Book:Tom M. Apostol/Mathematical Analysis

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## Tom M. Apostol:

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

Published $\text {1957}$, **Addison-Wesley**

### Subject Matter

### Contents

- Preface (January 1957)

- CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS
- 1-1 Introduction
- 1-2 Arithmetical properties of real numbers
- 1-3 Order properties of real numbers
- 1-4 Geometrical representation of real numbers
- 1-5 Decimal representation of real numbers
- 1-6 Rational numbers
- 1-7 Some irrational numbers
- 1-8 Some fundamental inequalities
- 1-9 Infimum and supremum
- 1-10 Complex numbers
- 1-11 Geometric representation of complex numbers
- 1-12 The imaginary unit
- 1-13 Absolute value of a complex number
- 1-14 Impossibility of ordering the complex numbers
- 1-15 Complex exponentials
- 1-16 The argument of a complex number
- 1-17 Integral pawers and roots of complex numbers
- 1-18 Complex logarithms
- 1-19 Complex powers
- 1-20 Complex sines and cosines

- CHAPTER 2. SOME BASIC NOTIONS OF SET THEORY
- 2-1 Fundamentals of set theory
- 2-2 Notations
- 2-3 Ordered pairs
- 2-4 Cartesian product of two sets
- 2-5 Relations and functions in the plane
- 2-6 General definition of relation
- 2-7 General definition of function
- 2-8 One-to-one functions and inverses
- 2-9 Composite functions
- 2-10 Sequences
- 2-11 The number of elements in a set
- 2-12 Set algebra

- CHAPTER 3. ELEMENTS OF POINT SET THEORY
- 3-1 Introduction
- 3-2 Intervals and open sets in $E_1$
- 3-3 The structure of open sets in $E_1$
- 3-4 Accumulation points and the Bolzano-Weierstrass theorem in $E_1$
- 3-5 Closed sets in $E_1$
- 3-6 Extensions to higher dimensions
- 3-7 The Heine-Borel covering theorem
- 3-8 Compactness
- 3-9 Infinity in the real number system
- 3-10 Infinity in the complex plane

- CHAPTER 4. THE LIMIT CONCEPT AND CONTINUITY
- 4-1 The definition of limit
- 4-2 Solne basic theorems on limits
- 4-3 The Cauchy condition
- 4-4 Algebra of limits
- 4-5 Continuity
- 4-6 Examples of continuous functions
- 4-7 Functions continuous on open or closed sets
- 4-8 Functions continuous on compact sets
- 4-9 Topological mappings
- 4-10 Properties of real-valued continuous functions
- 4-11 Uniform continuity
- 4-12 Discontinuities of real-valued functions
- 4-13 Monotonic functions
- 4-14 Necessary and sufficient conditions for continuity

- CHAPTER 5. DIFFERENTIATION OF FUNCTIONS OF ONE REAL VARIABLE
- 5-1 Introduction
- 5-2 Definition of derivative
- 5-3 Algebra of derivatives
- 5-4 The chain rule
- 5-5 One-sided derivatives and infinite derivatives
- 5-6 Functions with nonzero derivative
- 5-7 Functions with zero derivative
- 5-8 Rolle's theorem
- 5-9 The Mean Value Theorem of differential calculus
- 5-10 Intermediate value theorem for derivatives
- 5-11 Taylor's formula with remainder

- CHAPTER 6. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES
- 6-1 Introduction
- 6-2 The directional derivative
- 6-3 differentials of functions of one real variable
- 6-4 differentials of functions of several variables
- 6-5 The gradient vector
- 6-6 Differentials of composite functions and the chain rule
- 6-7 Cauchy's invariant rule
- 6-8 The Mean Value Theorem for functions of several variables
- 6-9 A sufficient condition for existence of the differential
- 6-10 Partial derivatives of higher order
- 6-11 Taylor's formula for functions of several variables
- 6-12 Differentiation of functions of a complex variable
- 6-13 The Cauchy-Riemann equations

- CHAPTER 7. APPLICATIONS OF PARTIAL DIFFERENTIATION
- 7-1 Introduction
- 7-2 Jacobians
- 7-3 Functions with nonzero Jacobian
- 7-4 The inverse function theorem
- 7-5 The implicit function theorem
- 7-6 Extremum problems
- 7-7 Sufficient conditions for a local extremum
- 7-8 Extremum problems with side conditions

- CHAPTER 8. FUNCTIONS OF BOUNDED VARIATION, RECTIFIABLE CURVES AND CONNECTED SETS
- 8-1 Introduction
- 8-2 Properties of monotonic functions
- 8-3 Functions of bounded variation
- 8-4 Total variation
- 8-5 Continuous functions of bounded variation
- 8-6 Curves
- 8-7 Equivalence of continuous vector-valued functions
- 8-8 Directed paths
- 8-9 Rectifiable curves
- 8-10 Properties of arc length
- 8-11 Connectedness
- 8-12 Components of a set
- 8-13 Regions
- 8-14 Statement of the Jordan curve theorem and related results

- CHAPTER 9. THEORY OF RIEMANN-STIELTJES INTEGRATION
- 9-1 Introduction
- 9-2 Notations
- 9-3 The definition of the Riemann-Stieltjes integral
- 9-4 Linearity properties
- 9-5 Integration by parts
- 9-6 Change of variable in a Riemann-Stieltjes integral
- 9-7 Reduction to a Riemann integral
- 9-8 Step functions as integrators
- 9-9 Monotonically increasing integrators. Upper and lower integrals
- 9-10 Riemann's condition
- 9-11 lntegrators of bounded variation
- 9-12 Sufficient conditions for existence of Riemann-Stieltjes integrals
- 9-13 Necessary conditions for existence of Riemann-Stieltjes integrals
- 9-14 Mean Value Theorems for Riemann-Stieltjes integrals
- 9-15 The integral as a function of the interval
- 9-16 Change of variable in a Riemann integral
- 9-17 Second Mean Value Theorem for Riemann integrals
- 9-18 Riemann-Stieltjes integrals depending on a parameter
- 9-19 Differentiation under the integral sign
- 9-20 Interchanging the order of integration
- 9-21 Oscillation of a function
- 9-22 Jordan content of bounded sets in $E_1$
- 9-23 A necessary and sufficient condition for integrabllity in terms of content
- 9-24 Outer Lebesgue measure of subsets of $E_1$
- 9-25 A necessary and sufficient condition for integrabllity in terms of measure
- 9-26 Complex-valued Riemann-Stieltjes integrals
- 9-27 Contour integrals
- 9-28 The winding number
- 9-29 Orientation of rectifiable Jordan curves
- 9-30 Addendum: Some theorems on outer Lebesgue measure

- CHAPTER 10. MULTIPLE INTEGRALS AND LINE INTEGRALS
- 10-1 Introduction
- 10-2 The measure (or content) of elementary sets in $E_n$
- 10-3 Riemann integration of bounded functions defined on intervals in $E_n$
- 10-4 Jordan content of bounded sets in $E_n$
- 10-5 Necessary and sufficient conditions for the existence of multiple integrals
- 10-6 Evaluation of a multiple integral by repeated integration
- 10-7 Multiple integration over more general sets
- 10-8 Mean Value Theorem for multiple integrals
- 10-9 Change of variable in a multiple integral
- 10-10 Line integrals
- 10-11 Line integrals with respect to arc length
- 10-12 The line integral of a gradient
- 10-13 Green's theorem for rectangles
- 10-14 Green's theorem for regions bounded by rectifiable Jordan curves
- 10-15 Independence of the path

- CHAPTER 11. VECTOR ANALYSIS
- 11-1 Introduction
- 11-2 Linear independence and bases in $E_n$
- 11-3 Geometric representation of vectors in $E_3$
- 11-4 Geometric interpretation of the dot product in $E_3$
- 11-5 The cross product of vectors in $E_3$
- 11-6 The scalar triple product
- 11-7 Derivatives of vector-valued functions
- 11-8 Elementary differential geometry of space curves
- 11-9 The tangent vector of a curve
- 11-10 Normal vectors, curvature, torsion
- 11-11 Vector fields
- 11-12 The gradient field in $E_n$
- 11-13 The curl of a vector field in $E_3$
- 11-14 The divergence of a vector field in $E_n$
- 11-15 The Laplacian operator
- 11-16 Surfaces
- 11-17 Explicit representation of a parametric surface
- 11-18 Area of a parametric surface
- 11-19 The sum of parametric surfaces
- 11-20 Surface integrals
- 11-21 The theorem of Stokes
- 11-22 Orientation of surfaces
- 11-23 Gauss' theorem (the divergence theorem)
- 11-24 Coordinate transformations

- CHAPTER 12. INFINITE SERIES AND INFINITE PRODUCTS
- 12-1 Introduction
- 12-2 Convergent and divergent sequences
- 12-3 Limit superior and limit inferior of a real-valued sequence
- 12-1 Monotonic sequences of real numbers
- 12-5 Infinite series
- 12-6 Inserting and removing parentheses
- 12-7 Alternating series
- 12-8 Absolute and conditional convergence
- 12-9 Real and imaginary parts of a complex series
- 12-10 Tests for convergence of series with positive terms
- 12-11 The ratio test and the root test
- 12-12 Dirichlet's test and Abel's test
- 12-13 Rearrangements of series
- 12-14 Double sequences
- 12-15 Double series
- 12-16 Multiplication of series
- 12-17 Cesàro summability
- 12-18 Infinite products

- CHAPTER 13. SEQUENCES OF FUNCTIONS
- 13-1 Introduction
- 13-2 Examples of sequences of real-valued functions
- 13-3 Definition of uniform convergence
- 13-4 An application to double sequences
- 13-5 Uniform convergence and continuity
- 13-6 The Cauchy condition for uniform convergence
- 13-7 Uniform convergence of infinite series
- 13-8 A space-filling curve
- 13-9 An application to repeated series
- 13-10 Uniform convergence and Riemann-Stieltjes integration
- 13-11 Uniform convergence and differentiation
- 13-12 Sufficient conditions for uniform convergence of a series
- 13-13 Bounded convergence. Arzelà's theorem
- 13-14 Mean convergence
- 13-15 Power series
- 13-16 Multiplication of power series
- 13-17 The substitution theorem
- 13-18 Real power series
- 13-19 Bernstein's theorem
- 13-20 The binomial series
- 13-21 Abel's limit theorem
- 13-22 Tauber's theorem

- CHAPTER 14. IMPROPER RIEMANN-STIELTZES INTEGRALS
- 14-1 Introduction
- 14-2 Infinite Riemann-Stieltjes integrals
- 14-3 Tests for convergence of infinite integrals
- 14-4 Infinite series and infinite integrals
- 14-5 Improper integrals of the second kind
- 14-6 Uniform convergence of improper integrals
- 14-7 Properties of functions defined by improper integrals
- 14-8 Repeated improper integrals
- 14-9 Integration of infinite series when improper integrals are involved

- CHAPTER 15. FOURIER SERIES AND FOURIER INTEGRALS
- 15-1 Introduction
- 15-2 Orthogonal systems on functions
- 15-3 Fourier series of a function relative to an orthonormal system
- 15-4 Mean-square approximation
- 15-5 Trigonometric Fourier series
- 15-6 The Riemann-Lebesgue lemma
- 15-7 Absolutely integrable functions
- 15-8 The Dirichlet integrals
- 15-9 An integral representation for the partial sums of a Fourier series
- 15-10 Riemann's localization theorem
- 15-11 Sufficient conditions for convergence of a Fourier series
- 15-12 Cesàro summability of Fourier series
- 15-13 Consequences of Fejér's theorem
- 15-14 Other forms of Fourier series
- 15-15 The Fourier integral theorem
- 15-16 The exponential form of the Fourier integral theorem
- 15-17 Integral transforms
- 15-18 Convolutions
- 15-19 The convolution theorem for Fourier transforms
- 15-20 The Laplace transform
- 15-21 The inversion formula for Laplace transforms

- CHAPTER 16. CAUCHY'S THEOREM AND RESIDUE CALCULUS
- 16-1 Analytic functions
- 16-2 The Cauchy integral theorem
- 16-3 Deformation of the contour
- 16-4 Cauchy's integral formula
- 16-5 The mean value of an analytic function on a circle
- 16-6 Cauchy's integral formula for the derivative of an analytic function
- 16-7 The existence of higher derivatives of an analytic function
- 16-8 Power series expansions for analytic functions
- 16-9 Zeros of analytic functions
- 16-10 The identity theorem for analytic functions
- 16-11 Laurent expansions for functions analytic on an annulus
- 16-12 Isolated singularities
- 16-13 The residue of a function at an isolated singular point
- 16-14 The Cauchy residue theorem
- 16-15 The difference between the number of zeros and the number of poles inside a closed contour
- 16-16 Evaluation of real-valued integrals by means of residues
- 16-17 Application of the residue theorem to the inversion formula for Laplace transforms
- 16-18 One-to-one analytic functions
- 16-19 Conformal mappings

- INDEX OF SPECIAL SYMBOLS

- INDEX

## Cited by

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces* - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach* - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.)

## Further Editions

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers

## Source work progress

- 1957: Tom M. Apostol:
*Mathematical Analysis*... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: $\text{1-4}$: Geometrical representation of real numbers