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Mathematical Analysis

Tom M. Apostol

• Chapter 1. The Real and Complex Number Systems
• 1.1 Introduction
• 1.2 The field axioms
• 1.3 The order axioms
• 1.4 Geometric representation of real numbers
• 1.5 Intervals
• 1.6 Integers
• 1.7 The unique factorization theorem for integers
• 1.8 Rational numbers
• 1.9 Irrational numbers
• 1.10 Upper bounds, maximum element, least upper bound (supremum)
• 1.11 The completeness axiom
• 1.12 Some properties of the supremum
• 1.13 Properties of the integers deduced from the completeness axiom
• 1.14 The Archimedean property of the real-number system
• 1.15 Rational numbers with finite decimal representation
• 1.16 Finite decimal approximations to real numbers
• 1.17 Infinite decimal representation of real numbers
• 1.18 Absolute values and the triangle inequality
• 1.19 The Cauchy-Schwarz inequality
• 1.20 Plus and minus infinity and the extended real number system R*
• 1.21 Complex numbers
• 1.22 Geometric representation of complex numbers
• 1.23 The imaginary unit
• 1.24 Absolute value of a complex number
• 1.25 Impossibility of ordering the complex numbers
• 1.26 Complex exponentials
• 1.27 Further properties of complex exponentials
• 1.28 The argument of a complex number
• 1.29 Integral powers and roots of complex numbers
• 1.30 Complex logarithms
• 1.31 Complex powers
• 1.32 Complex sines and cosines
• 1.33 Infinity and the extended complex plane C*
• Exercises
• Chapter 2. Some Basic Notions of Set Theory
• 2.1 Introduction
• 2.2 Notations
• 2.3 Ordered pairs
• 2.4 Cartesian product of two sets
• 2.5 Relations and functions
• 2.6 Further terminology concerning functions
• 2.7 One-to-one functions and inverses
• 2.8 Composite functions
• 2.9 Sequences
• 2.10 Similar (equinumerous) sets
• 2.11 Finite and infinite sets
• 2.12 Countable and uncountable sets
• 2.13 Uncountability of the real-number system
• 2.14 Set algebra
• 2.15 Countable collections of countable sets
• Exercises
• Chapter 3. Elements of Point Set Topology
• 3.1 Introduction
• 3.2 Euclidean space Rn
• 3.3 Open balls and open sets in Rn
• 3.4 The structure of open sets in R1
• 3.5 Closed sets
• 3.6 Adherent points. Accumulation points
• 3.7 Closed sets and adherent points
• 3.8 The Bolzano-Weierstrass theorem
• 3.9 The Cantor intersection theorem
• 3.10 The Lindelöf covering theorem
• 3.11 The Heine-Borel covering theorem
• 3.12 Compactness in Rn
• 3.13 Metric spaces
• 3.14 Point set topology in metric spaces
• 3.15 Compact subsets of a metric space
• 3.16 Boundary of a set
• Exercises
• Chapter 4. Limits and Continuity
• 4.1 Introduction
• 4.2 Convergent sequences in a metric space
• 4.3 Cauchy sequences
• 4.4 Complete metric spaces
• 4.5 Limit of a function
• 4.6 Limits of complex-valued functions
• 4.7 Limits of vector-valued functions
• 4.8 Continuous functions
• 4.9 Continuity of composite functions
• 4.10 Continuous complex-valued and vector-valued functions
• 4.11 Examples of continuous functions
• 4.12 Continuity and inverse images of open or closed sets
• 4.13 Functions continuous on compact sets
• 4.14 Topological mappings (homeomorphisms)
• 4.15 Bolzano’s theorem
• 4.16 Connectedness
• 4.17 Components of a metric space
• 4.18 Arcwise connectedness
• 4.19 Uniform continuity
• 4.20 Uniform continuity and compact sets
• 4.21 Fixed-point theorem for contractions
• 4.22 Discontinuities of real-valued functions
• 4.23 Monotonic functions
• Exercises
• Chapter 5. Derivatives
• 5.1 Introduction
• 5.2 Definition of derivative
• 5.3 Derivatives and continuity
• 5.4 Algebra of derivatives
• 5.5 The chain rule
• 5.6 One-sided derivatives and infinite derivatives
• 5.7 Functions with nonzero derivative
• 5.8 Zero derivatives and local extrema
• 5.9 Rolle’s theorem
• 5.10 The Mean-Value Theorem for derivatives
• 5.11 Intermediate-value theorem for derivatives
• 5.12 Taylor’s formula with remainder
• 5.13 Derivatives of vector-valued functions
• 5.14 Partial derivatives
• 5.15 Differentiation of functions of a complex variable
• 5.16 The Cauchy-Riemann equations
• Exercises
• 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 [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
• 7.1 Introduction
• 7.2 Notation
• 7.3 The definition of the Riemann-Stieltjes integral
• 7.4 Linear properties
• 7.5 Integration by parts
• 7.6 Change of variable in a Riemann-Stieltjes integral
• 7.7 Reduction to a Riemann integral
• 7.8 Step functions as integrators
• 7.9 Reduction of a Riemann-Stieltjes integral to a finite sum
• 7.10 Euler’s summation formula
• 7.11 Monotonically increasing integrators. Upper and lower integrals
• 7.12 Additive and linearity properties of upper and lower integrals
• 7.13 Riemann’s condition
• 7.14 Comparison theorems
• 7.15 Integrators of bounded variation
• 7.16 Sufficient conditions for existence of Riemann-Stieltjes integrals
• 7.17 Necessary conditions for existence of Riemann-Stieltjes integrals
• 7.18 Mean Value Theorems for Riemann-Stieltjes integrals
• 7.19 The integral as a function of the interval
• 7.20 Second fundamental theorem of integral calculus
• 7.21 Change of variable in a Riemann integral
• 7.22 Second Mean-Value Theorem for Riemann integrals
• 7.23 Riemann-Stieltjes integrals depending on a parameter
• 7.24 Differentiation under the integral sign
• 7.25 Interchanging the order of integration
• 7.26 Lebesgue’s criterion for existence of Riemann integrals
• 7.27 Complex-valued Riemann-Stieltjes integrals
• Exercises
• Chapter 8. Infinite Series and Infinite Products
• 8.1 Introduction
• 8.2 Convergent and divergent sequences of complex numbers
• 8.3 Limit superior and limit inferior of a real-valued sequence
• 8.4 Monotonic sequences of real numbers
• 8.5 Infinite series
• 8.6 Inserting and removing parentheses
• 8.7 Alternating series
• 8.8 Absolute and conditional convergence
• 8.9 Real and imaginary parts of a complex series
• 8.10 Tests for convergence of series with positive terms
• 8.11 The geometric series
• 8.12 The integral test
• 8.13 The big oh and little oh notation
• 8.14 The ratio test and the root test
• 8.15 Dirichlet’s test and Abel’s test
• 8.16 Partial sums of the geometric series Σ zn on the unit circle |z| = 1
• 8.17 Rearrangements of series
• 8.18 Riemann’s theorem on conditionally convergent series
• 8.19 Subseries
• 8.20 Double sequences
• 8.21 Double series
• 8.22 Rearrangement theorem for double series
• 8.23 A sufficient condition for equality of iterated series
• 8.24 Multiplication of series
• 8.25 Cesàro summability
• 8.26 Infinite products
• 8.27 Euler’s product for the Riemann zeta function
• Exercises
• 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
• 10.1 Introduction
• 10.2 The integral of a step function
• 10.3 Monotonic sequences of step functions
• 10.4 Upper functions and their integrals
• 10.5 Riemann-integrable functions as examples of upper functions
• 10.6 The class of Lebesgue-integrable functions on a general interval
• 10.7 Basic properties of the Lebesgue integral
• 10.8 Lebesgue integration and sets of measure zero
• 10.9 The Levi monotone convergence theorems
• 10.10 The Lebesgue dominated convergence theorem
• 10.11 Applications of Lebesgue’s dominated convergence theorem
• 10.12 Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals
• 10.13 Improper Riemann integrals
• 10.14 Measurable functions
• 10.15 Continuity of functions defined by Lebesgue integrals
• 10.16 Differentiation under the integral sign
• 10.17 Interchanging the order of integration
• 10.18 Measurable sets on the real line
• 10.19 The Lebesgue integral over arbitrary subsets of R
• 10.20 Lebesgue integrals of complex-valued functions
• 10.21 Inner products and norms
• 10.22 The set L2(I) of square-integrable functions
• 10.23 The set L2(I) as a semimetric space
• 10.24 A convergence theorem for series of functions in L2(I)
• 10.25 The Riesz-Fischer theorem
• Exercises
• Chapter 11. Fourier Series and Fourier Integrals
• 11.1 Introduction
• 11.2 Orthogonal systems of functions
• 11.3 The theorem on best approximation
• 11.4 The Fourier series of a function relative to an orthonormal system
• 11.5 Properties of the Fourier coefficients
• 11.6 The Riesz-Fischer theorem
• 11.7 The convergence and representation problems for trigonometric series
• 11.8 The Riemann-Lebesgue lemma
• 11.9 The Dirichlet integrals
• 11.10 An integral representation for the partial sums of a Fourier series
• 11.11 Riemann’s localization theorem
• 11.12 Sufficient conditions for convergence of a Fourier series at a particular point
• 11.13 Cesàro summability of Fourier series
• 11.14 Consequences of Fejér’s theorem
• 11.15 The Weierstrass approximation theorem
• 11.16 Other forms of Fourier series
• 11.17 The Fourier integral theorem
• 11.18 The exponential form of the Fourier integral theorem
• 11.19 Integral transforms
• 11.20 Convolutions
• 11.21 The convolution theorem for Fourier transforms
• 11.22 The Poisson summation formula
• Exercises
• Chapter 12. Multivariable Differential Calculus
• 12.1 Introduction
• 12.2 The directional derivative
• 12.3 Directional derivatives and continuity
• 12.4 The total derivative
• 12.5 The total derivative expressed in terms of partial derivatives
• 12.6 An application to complex-valued functions
• 12.7 The matrix of a linear function
• 12.8 The Jacobian matrix
• 12.9 The chain rule
• 12.10 Matrix form of the chain rule
• 12.11 The Mean-Value Theorem for differentiable functions
• 12.12 A sufficient condition for differentiability
• 12.13 A sufficient condition for equality of mixed partial derivatives
• 12.14 Taylor’s formula for functions from Rn to R1
• Exercises
• Chapter 13. Implicit Functions and Extremum Problems
• 13.1 Introduction
• 13.2 Functions with nonzero Jacobian determinant
• 13.3 The inverse function theorem
• 13.4 The implicit function theorem
• 13.5 Extrema of real-valued functions of one variable
• 13.6 Extrema of real-valued functions of several variables
• 13.7 Extremum problems with side conditions
• Exercises
• Chapter 14. Multiple Riemann Integrals
• 14.1 Introduction
• 14.2 The measure of a bounded interval in Rn
• 14.3 The Riemann integral of a bounded function defined on a compact interval in Rn
• 14.4 Sets of measure zero and Lebesgue’s criterion for existence of a multiple Riemann integral
• 14.5 Evaluation of a multiple integral by iterated integration
• 14.6 Jordan-measurable sets in Rn
• 14.7 Multiple integration over Jordan-measurable sets
• 14.8 Jordan content expressed as a Riemann integral
• 14.9 Additive property of the Riemann integral
• 14.10 Mean-Value Theorem for multiple integrals
• Exercises
• Chapter 15. Multiple Lebesgue Integrals
• 15.1 Introduction
• 15.2 Step functions and their integrals
• 15.3 Upper functions and Lebesgue-integrable functions
• 15.4 Measurable functions and measurable sets in R
• 15.5 Fubini’s reduction theorem for the double integral of a step function
• 15.6 Some properties of sets of measure zero
• 15.7 Fubini’s reduction theorem for double integrals
• 15.8 The Tonelli-Hobson test for integrability
• 15.9 Coordinate transformations
• 15.10 The transformation formula for multiple integrals
• 15.11 Proof of the transformation formula for linear coordinate transformations
• 15.12 Proof of the transformation formula for the characteristic function of compact cube
• 15.13 Completion of the proof of the transformation formula
• Exercises
• Chapter 16. Cauchy’s Theorem and the Residue Calculus
• 16.1 Analytic functions
• 16.2 Paths and curves in the complex plane
• 16.3 Contour integrals
• 16.4 The integral along a circular path as a function of the radius
• 16.5 Cauchy’s integral theorem for a circle
• 16.6 Homotopic curves
• 16.7 Invariance of contour integrals under homotopy
• 16.8 General form of Cauchy’s integral theorem
• 16.9 Cauchy’s integral formula
• 16.10 The winding number of a circuit with respect to a point
• 16.11 The unboundedness of the set of points with winding number zero
• 16.12 Analytic functions defined by contour integrals
• 16.13 Power-series expansions for analytic functions
• 16.14 Cauchy’s inequalities. Liouville’s theorem
• 16.15 Isolation of the zeros of an analytic function
• 16.16 The identity theorem for analytic functions
• 16.17 The maximum and minimum modulus of an analytic function
• 16.18 The open mapping theorem
• 16.19 Laurent expansions for functions analytic in an annulus
• 16.20 Isolated singularities
• 16.21 The residue of a function at an isolated singular point
• 16.22 The Cauchy residue theorem
• 16.23 Counting zeros and poles in a region
• 16.24 Evaluation of real-valued integrals by means of residues
• 16.25 Evaluation of Gauss’s sum by residue calculus
• 16.26 Application of the residue theorem to the inversion formula for Laplace transforms
• 16.27 Conformal mappings
• Exercises
• Index of Special Symbols
• Index