# Book:Tom M. Apostol/Mathematical Analysis

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

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

### 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
6-6 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

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