数学分析（英文版·原书第2版·典藏版） 机械工业出版社 9787111706106 (美)汤姆·M.阿波斯托尔
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作者(美)汤姆·M.阿波斯托尔
出版社机械工业出版社
ISBN9787111706106
出版时间2022-06
装帧其他
开本16开
定价139元
货号31478072
上书时间2024-06-21
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作者简介
汤姆·M. 阿波斯托尔(Tom M. Apostol)是加州理工学院数学系荣誉教授。他于1946年在华盛顿大学西雅图分校获得数学硕士学位，于1948年在加州大学伯克利分校获得数学博士学位。
目录
Chapter 1 The Real and Complex Number Systems<br/>1.1 Introduction 1<br/>1.2 The field axioms . 1<br/>1.3 The order axioms 2<br/>1.4 Geometric representation of real numbers 3<br/>1.5 Intervals 3<br/>1.6 Integers 4<br/>1.7 The unique factorization theorem for integers 4<br/>1.8 Rational numbers 6<br/>1.9 Irrational numbers 7<br/>1.10 Upper bounds, maximum element, least upper bound(supremum) . 8<br/>1.11 The completeness axiom 9<br/>1.12 Some properties of the supremum 9<br/>1.13 Properties of the integers deduced from the completeness axiom 10<br/>1.14 The Archimedean property of the real-number system . 10<br/>1.15 Rational numbers with finite decimal representation 11<br/>1.16 Finite decimal approximations to real numbers 11<br/>1.17 Infinite decimal representation of real numbers . 12<br/>1.18 Absolute values and the triangle inequality 12<br/>1.19 The Cauchy—Schwarz inequality 13<br/>1.20 Plus and minus infinity and the extended real number system R* 14<br/>1.21 Complex numbers 15<br/>1.22 Geometric representation of complex numbers 17<br/>1.23 The imaginary unit 18<br/>1.24 Absolute value of a complex number . 18<br/>1.25 Impossibility of ordering the complex numbers . 19<br/>1.26 Complex exponentials 19<br/>1.27 Further properties of complex exponentials 20<br/>1.28 The argument of a complex number . 20<br/>1.29 Integral powers and roots of complex numbers . 21<br/>1.30 Complex logarithms 22<br/>1.31 Complex powers 23<br/>1.32 Complex sines and cosines 24<br/>1.33 Infinity and the extended complex plane C* 24<br/>Exercises 25<br/>Chapter 2 Some Basic Notions of Set Theory<br/>2.1 Introductiou 32<br/>2.2 Notations 32<br/>2.3 Ordered pairs 33<br/>2.4 Cartesian product of two sets 33<br/>2.5 Relations and functions 34<br/>2.6 Further terminology concerning functions 35<br/>2.7 One-to-one functions and inverses 36<br/>2.8 Composite functions 37<br/>2.9 Sequences. 38<br/>2.10 Similar (equinumerous) sets 38<br/>2.11 Finite and infinite sets 39<br/>2.12 Countable and uncountable sets 39<br/>2.13 Uncountability of the real-number system 42<br/>2.14 Set algebra 43<br/>2.15 Countable collections of countable sets <br/>Exercises 43<br/>Chapter 3 Elements of Point Set Topology<br/>3.1 Introduction 47<br/>3.2 Euclidean space R't 47<br/>3.3 Open balls and open sets in R* 49<br/>3.4 The structure of open sets in RH 50<br/>3.5 Closed sets . 52<br/>3.6 Adhèrent points. Accumulation points 52<br/>3.7 Closed sets and adhèrent points 53<br/>3.8 The Bolzano—Weierstrass theorem 54<br/>3.9 The Cantor intersection theorem 56<br/>3.10 The Lindel?f covering theorem 56<br/>3.11 The Heine—Borel covering theorem 58<br/>3.12 Compactness in R‘ 59<br/>3.13 Metric spaces 60<br/>3.14 Point set topology in metric spaces 61<br/>3.15 Compact subsets of a metric space 63<br/>3.16 Boundary of a set<br/>Exercises 65<br/>Chaqter 4 Limits and Continuity<br/> 4.1 Introduction 70<br/>4.2 Convergent sequences in a metric space 72<br/>4.3 Cauchy sequences 74<br/>4.4 Complete metric spaces . 74<br/>4.5 Limit of a function 76<br/>4.6 Limits of complex-valued functions<br/> 4.7 Limits of vector-valued functions 77<br/>4.8 Continuous functions 78<br/>4.9 Continuity of composite functions.<br/>4.10 Continuous complex-valued and vector-valued functions 79<br/>4.11 Examples of continuous functions 80<br/>4.12 Continuity and inverse images of open or closed sets 80<br/>4.13 Functions continuous on compact sets 81<br/>4.14 Topolo$ical mappings (homeomorphisms) 82<br/>4.15 Bolzano’s theorem 84<br/>4.16 Connectedness 84<br/>4.17 Components of a metric space . 86<br/>4.18 Arcwise connectedness 87<br/>4.19 Uniform continuity 88<br/>4.20 Uniform continuity and compact sets 90<br/>4.21 Fixed-point theorem for contractions 91<br/>4.22 Discontinuities of real-valued functions 92<br/>4.23 Monotonic functions 94<br/>Exercises 95<br/>Chapter 5 DerJvatives<br/> 5.1Introduction 104<br/>5.2 Definition of derivative .104<br/>5.3 Derivatives and continuity 105<br/>5.4 Algebra of derivatives106<br/>5.5 The chain rule 106<br/>5.6 One-sided derivatives and infinite derivatives 106<br/>5.7 Functions with nonzero derivative 108<br/>5.8 Zero derivatives and local extrema 109<br/>5.9 Rolle’s theorem 110<br/>5.10 The Mean-Value Theorem for derivatives 110<br/>5.11 Intermediate-value theorem for derivatives 111<br/>5.12 Taylor’s formula with remainder 113<br/>5.13 Derivatives of vector-valued functions 114<br/>5.14 Partial derivatives 115<br/>5.15 DiPerentiation of functions of a complex variable 116<br/>5.16 The Cauchy Riemann equations 118<br/>Exercises 121<br/>Chapter 6 Functions of Bounded Variation and Reetifiable Curves<br/>6.1 Introduction 127<br/>6.2 PropertleS Of monotonic functions 128<br/>6.3 Functions of bounded variation 129<br/>6.4 Total variation 130<br/>6.5 Additive property of total variation 131<br/>6.6 Total variation on (a, x) as a function of x 132<br/>6.7 Functions of bounded variation expressed as the diPerence of increasing functions . 133<br/>6.8 Continuous functions of bounded variation132<br/>6,9 Curves and paths133<br/>6.10 Rectifiable paths and arc length134<br/>6.11 Additive and continuity properties of arc length135<br/>6.12 Equivalence of paths. Change of parameter136<br/>Exercises136<br/>Chapter 7The Riemann—Stieltjes Integral<br/>7.1 Introduction140<br/>7.2 Notation 141<br/>7.3 The definition of the Riemann—Stieltjes integral 141<br/>7.4 Linear properties<br/>7.5 Integration by parts . 144<br/>7.6 Change of variable in a Riemann Stieltjes integral 144<br/>7.7 Reduction to a Riemann integral 145<br/>7.8 Step functions as integrators 147<br/>7.9 Reduction of a Riemann—Stieltjes integral to a finite sum 148<br/>7.10 Euler’s summation formula 149<br/>7.11 Monotonically increasing integrators.Upper and lower integrals .150<br/>7.12 Additive and linearity properties of upper and lower integrals153<br/>7.13 Riemann’s condition 153<br/>7.14 Comparison theorems 155<br/>7.15 Integrators of bounded variation 156<br/>7.16 Sufficient conditions for existence of Riemann—Stieltjes integrals159<br/>7.17 Necessary conditions for existence of Riemann—Stieltjes integrals 160<br/>7.18 Mean Value Theorems for Riemann—Stieltjes integrals 160<br/>7.19 The integral as a function of the interval . 161<br/>7.20 Second fundamental theorem of integral calculus162<br/>7.21 Change of variable in a Riemann integral 163<br/>7.22 Second Mean-Value Theorem for Riemann integrals165<br/>7.23 Riemann—Stieltjes integrals depending on a parameter 166<br/>7.24 Dikerentiation under the integral sign 167<br/>7.25 Interchanging the order of integration 167<br/>7.26 Lebesgue’s criterion for existence of Riemann integrals169<br/>7.27 Complex-valued Riemann—Stieltjes integrals 173<br/>Exercises 174<br/>Chapter 8 Infinite Series and Infinite Products<br/>8.1 Introduction 183<br/>8.2 Convergent and divergent sequences of complex numbers 183<br/>8.3 Limit superior and limit inferior of a real-valued sequence 184<br/>8.4 Monotonic sequences of real numbers 185<br/>8.5 Infinite series 185<br/>8.6 Inserting and removing parenthèses 187<br/>8.7 Alternating series 188<br/>8.8 Absolute and conditional convergence 189<br/>8.9 Real and imaginary parts of a complex series 189<br/>8.10 Tests for convergence of series with positive terms 190<br/>8.11 The geometric series 190<br/>8.12 The integral test 191<br/>8.13 The big oh and little oh notation 192<br/>8.14 The ratio test and the root test 193<br/>8.15 Dirichlet’s test and Abel’s test 193<br/>8.16 Partial sums of the geometric series Z z‘ on the unit circle ]z] 195<br/>8.17 Rearrangements of series 196<br/>8.18 Riemann’s theorem on conditionally convergent series 197<br/>8.19 Subseries 197<br/>8.20 Double sequences 199<br/>8.21 Double series 200<br/>8.22 Rearran$ement theorem for double series 201<br/>8.23 A sufficient condition for equality of iterated series 202<br/>8.24 Multiplication of series 203<br/>8.25 Cesàro summability 205<br/>8.26 Infinite products
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