Edition:

2

Publisher:

Addison Wesley

Number of Pages:

492

Price:

73.33

ISBN:

9780201002881

This is a very carefully written introduction to real analysis. When Apostol published the first edition in 1957, he intended it to be intermediate between calculus and real variables theory, and it still has a strong feeling of being a transitional course. It starts out with several chapters on the number line and point set topology, then proves all the basic facts that are taken for granted in differential calculus courses. It then proceeds into what is new material for most students, with two new theories of integration (Riemann-Stieltjes and Lebesgue), multivariable and vector calculus (focusing on existence theorems such as the Implicit Function Theorem rather than physical applications), some advanced theorems in sequences and series, approximation by sequences of functions and orthonormal bases, and a brief introduction to complex variables.

The book crams a lot of material into a modest number of pages, though without feeling rushed. This is done primarily by sticking to the main threads of the subject in the exposition, while moving a lot of related and more specialized results to the exercises. The proofs themselves are not overly brief, although there’s not much handholding and only a few examples are given, so (a) you have to pay attention, and (b) it’s very helpful to have already taken advanced calculus so that you can orient yourself. In many ways this book resembles the British analysis books of the early twentieth century, such as Hardy’s A Course of Pure Mathematics and Titchmarsh’s Theory of Functions. The approach is generally more modern, and there are many more exercises, but it has the same kind of concision.

The book makes a good balance between simplicity and generality. For example, the Riemann-Stieltjes integral rather than the plain Riemann integral is used for the elementary integration (before Lebesgue). It’s not any harder, introduces some other valuable concepts such as functions of bounded variation, and gives us a tool which is often useful in discrete and discontinuous problems. For another example, Fourier series are developed first for orthogonal systems and then specialized for trigonometric series. Again, this is not any harder and gives us a better insight into why Fourier series work.

Rudin’s Principles of Mathematical Analysis is the one to beat in this field. Apostol’s treatment is not that different from Rudin’s. The books were written about the same time, with Rudin having editions in 1953, 1964, and 1976, and Apostol in 1957 and 1974. The coverage of the two books is roughly similar. Rudin is slightly more abstract and slanted more toward multivariable analysis. Both have concise proofs, a shortage of examples, and numerous challenging exercises. Both cover the Riemann-Stieltjes integral rather than the plain Riemann integral. Both cover the Lebesgue integral, although Rudin is more skimpy and uses the conventional measure theory approach, while Apostol follows Riesz & Sz.-Nagy et al., using a functional-analysis approach through step functions.

Apostol has taken care to modularize his book, so that it can be used for several different courses and the material studied in different orders. Rudin’s treatment is more tightly integrated. This often makes Apostol easier to use as a reference, because everything you need to understand a theorem will be close by.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

Date Received:

Friday, July 24, 2009

Reviewable:

Yes

Publication Date:

1974

Format:

Paperback

Audience:

Category:

Textbook

Allen Stenger

11/25/2012

- 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
**R**^{n} - 3.3 Open balls and open sets in
**R**^{n} - 3.4 The structure of open sets in
**R**^{1} - 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
**R**^{n} - 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
*z*on the unit circle |^{n}*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
*L*^{2}(*I*) of square-integrable functions - 10.23 The set
*L*^{2}(*I*) as a semimetric space - 10.24 A convergence theorem for series of functions in
*L*^{2}(*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
**R**to^{n}**R**^{1} - 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
**R**^{n} - 14.3 The Riemann integral of a bounded function defined on a compact interval in
**R**^{n} - 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
**R**^{n} - 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

- 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
**R**^{n} - 3.3 Open balls and open sets in
**R**^{n} - 3.4 The structure of open sets in
**R**^{1} - 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
**R**^{n} - 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
*z*on the unit circle |^{n}*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
*L*^{2}(*I*) of square-integrable functions - 10.23 The set
*L*^{2}(*I*) as a semimetric space - 10.24 A convergence theorem for series of functions in
*L*^{2}(*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
**R**to^{n}**R**^{1} - 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
**R**^{n} - 14.3 The Riemann integral of a bounded function defined on a compact interval in
**R**^{n} - 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
**R**^{n} - 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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