数学分析(英文版.第2版) 作者:阿波斯托尔

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书名:数学分析(英文版.第2版)
作者:阿波斯托尔
译者:
出版社:机械工业出版社
价格:49元

简介

本书是一部现代数学名著。自20世纪70年代面世以来，一直受到西方学术界、教育界的广泛推崇，被许多知名大学指定为教材。相比于同类书籍，它的特点在于：选取的论据更适子教学使用。论证详尽，可读性更强。习题丰富，覆盖各个方面、各级难度。可根据教学需要选用不同章节。

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 Rs
3.4 The structure of open sets in Rx
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 Lindelof covering theorem
3.11 The Heine-Borel covering theorem
3.12 Compactness in Rs
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 Coatinnity
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 Bolzanos theorem
4.16 Connectedness
4.17 Components of a metric space
4.18 Arcwise cormectedness
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 0ne-sided derivatives and infinite derivatives
5.7 Functions with nonzero derivative
5.8 Zero derivatives and local extrema
5.9 Rolles theorem
5.10 The Mean-Value Theorem for derivatives
5.11 Intermediate-value theorem for derivatives
5.12 Taylors formula with remainder
5.13 Derivatives of vector-valued functions
5.14 Partial derivatives
5.15 Differentiatiofi 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 Eulers summation formula
7.11 Monotonically increasing integrators. Upper and lower integrals
7.12 Additive and linearity properties of upper and lower integrals
7.13 Riemanns 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 Lebesgues 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 Dirichlets test and Abels test
8.16 Partial sums of the geometric series ∑zn on the unit circle [z] = 1
8.17 Rearrangements of series
8.18 Riemanns 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 Cesaro summability
8.26 Infinite products
8.27 Eulers product for the Riemarm 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 integratea term
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 Taylors series generated by a function
9.20 Bernsteins theorem
9.21 The binomial series
9.22 Abels limit theorem
9.23 Taubers 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 Lebesgues 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(1) 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 Riemanns localization theorem
11.12 Sufficient conditions for convergence of a Fourier series at a particular

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