Book:Tom M. Apostol/Mathematical Analysis

Subject Matter
Analysis

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



Source work progress
* : Chapter $1$: The Real and Complex Number Systems: $\text{1-2}$ Order properties of real numbers