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Mathematical Analysis

Table of Contents

• 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

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