音像数学分析(英文版·原书第2版·典藏版)(美)汤姆·M.阿波斯托尔

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 comleess axiom 9<br/>1.12 Some properties of the supremum 9<br/>1.13 Properties of the integers deduced from the comleess axim 0<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. The imaginary unit 18<br/>1.24 Absolute value of a complex number . 18<br/>1.25 Isibility 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 <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/>. 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 Coite 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 LimisndCntinuity<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 coite 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 continuiyndcmpact sets 90<br/>4.21 Fixed-point theorem for contractions 91<br/>4.22 Discontinuities of real-valued functions 92<br/>4. 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 variation 132<br/>6,9 Curves and paths 133<br/>6.10 Rectifiable paths and arc length 134<br/>6.11 Additive and continuity properties of arc length 135 <br/>6.12 Equivalence of paths. Change of parameter 136<br/>Exercises 136<br/>Chapter 7 The Riemann—Stieltjes Integral<br/>7.1 Introduction 140<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 integrals 153<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 integrals 159<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 calculus 162<br/>7.21 Change of variable in a Riemann integral 163<br/>7.22 Second Mean-Value Theorem for Riemann integrals 165<br/>7. 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 integrals 169<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 an eoving 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. 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 209<br/>8.27 Euler’s product for the Riemann zeta function 209<br/>Exercises 210<br/>Chaqter 9 Sequences of Functions<br/>9.1 Pointwise convergence of sequences of functions 218<br/>9.2 Examples of sequences of real-valued functions . 219<br/>9.3 Definition of uniform convergence 220<br/>9.4 Uniform convergence and continuity . 221<br/>9.5 The Cauchy condition for uniform convergence 222<br/>9.6 Uniform convergence of infinite series of functions . 2<br/>9.7 A space-filling curve 224<br/>9.8 Uniform convergence and Riemann—Stieltjes integration 225<br/>9.9 Nonuniformly convergent sequences that can be integrated term 226<br/>9.10 Uniform convergence and diPerentiation 228<br/>9.11 Sufficient conditions for uniform convergence of a series 0<br/>9.12 Uniform convergence and double sequences . 1<br/>9.13 Mean convergence 2<br/>9.14 Power series 4<br/>9.15 Multiplication of power series . <br/>9.16 The substitution theorem . <br/>9.17 Reciprocal of a power series <br/>9.18 Real power series 240<br/>9.19 The Taylor’s series generated by a function 241<br/>9.20 Bernstein’s theorem . 242<br/>9.21 The binomial series 244<br/>9.22 Abel’s limit theorem 246<br/>9. Tauber’s theorem 246<br/>Exercises 247<br/> Chapter 10 The Lebesgue Integral<br/>10.1 Introduction .. . 252<br/>10.2 The integral of a step function .. . 253<br/>10.3 Monotonic sequences of step functions . 254<br/>10.4 Upper functions and their integrals ... . 256<br/>10.5 Riemann-integrable functions as examples of upper functions 259<br/>10.6 The class of Lebesgue-integrable functions on a general internal 260<br/>10.7 Basic properties of the Lebesgue integral . ... 261<br/>10.8 Lebesgue integration and sets of measure zero . 264<br/>10.9 The Levi monotone convergence theorems . . 265<br/>10.10 The Lebesgue dominated convergence theorem . 270<br/>10.11 Applications of Lebesgue’s dominated convergence theorem 272<br/>10.12 Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals 274<br/>10.13 Improper Riemann integrals 276<br/>10.14 Measurable functions 279<br/>10.15 Continuity of functions defined by Lebesgue integrals 281<br/>10.16 DiPerentiation under the integral sign 283<br/>10.17 Interchanging the order of integration 287<br/>10.18 Measurable sets on the real line 289 <br/>10.19 The Lebesgue integral over arbitrary subsets of R 291<br/>10.20 Lebesgue integrals of complex-valued functions . 292<br/>10.21 Inner producsndnrms 293<br/>10.22 The set L2(/ ) of square-integrable functions . 294<br/>10. The set L2(J) as a semimetric space 295<br/>10.24 A convergence theorem for series of functions in L2(I) 295<br/>10.25 The Riesz—Fischer theorem 297<br/>Exercises 298<br/>Chapter 11 Fourier Series and Fourier Integrals<br/>11.1 Introduction . . .. 306<br/>11.2 Orthogonal systems of functions .. . 306<br/>11.3 The theorem on best approximation . ... 307<br/>11.4 The Fourier series of a function relative to an orthonormal system 309<br/>11.5 Properties of ihe Fourier coeificients . . . . 309<br/>11.6 The Riesz—Fischer theorem .. .... . 31i<br/>11.7 The convergence and representation problems for trigonometric series 312<br/>11.8 The Riemann—Lebesgue lemma<br/>11.9 The Dirichlet integrals<br/>11.10 An integral representation for the partial sums of a Fourier series 317<br/>11.11 Riemann’s localization theorem 318<br/>11.12 Sufficient conditions for convergence of a Fourier series at a particular point 319<br/>11.13 Cesàro summability of Fourier series 320<br/>11.14 Consequences of Fejér’s theorem 321<br/>11.15 The Weierstrass approximation theorem 322<br/>11.16 Other forms of Fourier series 322<br/>11.17 The Fourier integral theorem 3<br/>11.18 The exponential form of the Fourier integral theorem 325<br/>11.19 Integral transforms 327<br/>11.20 Convolution 327<br/>11.21 The convolution theorem for Fourier transforms 329<br/>11.22 The Poisson summation formula 332<br/>Exercise 335<br/>Chaqter 12 Sufficient conditions for convergence of a Fourier series at a particular point<br/>12.1 Introduction 344<br/>12.2 The directional derivative 344<br/>1. Directional derivatives and continuity 345<br/>12.4 The total derivative 346<br/>12.5 The total derivative expressed in terms of partial derivatives 347<br/>12.6 An application to complex-valued functions 348<br/>12.7 The matrix of a linear function 349<br/>12.8 The Jacobian matrix 351<br/>12.9 The chain rule 352<br/>12.10 Matrix form of the chain rule 353<br/>12.11 The Mean-Value Theorem for di#erentiable functions 355<br/>12.12 A sufficient condition for diPerentiability 357<br/>12.13 A sufficient condition for equality of mixed partial derivatives 358<br/>12.14 Taylor's formula for functions from R* to R' 361<br/> Exercises 362<br/>Chapter 13 Implicit Functions and Extremum Problems<br/>13.1 Introduction 367<br/>13.2 Functions with nonzero Jacobian determinant 368<br/>13.3 The inverse function theorem 372<br/>13.4 The implicit function theorem 373<br/>13.5 Extrema of real-valued functions of one variable 375<br/>13.6 Extrema of real-valued functions of several variables 376<br/>13.7 Extremum problems with side conditions 380<br/>Exercises 384<br/>Chapter 14 Multiple Riemam liitegrals<br/>14.1 Introduction 388<br/>14.2 The measure of a bounded internal in R* 388<br/> 14.3 The Riemann integral of a bounded function defined on a compact 389<br/>interval in R* <br/>14.4 Sets of measure zero and kebesgue’s criterion for existence of a multiple<br/>Riemann integral391<br/>14.5 Evaluation of a mutl iteral by iterated integration 391<br/>14.6 Jordan-measurable sets in R* 396<br/>14.7 Mutl iteration over Jordan-measurable sets 397<br/>14.8 Jordan content expressed as a Riemann integral 398<br/>14.9 Additive property of the Riemann integral 366<br/>14.10 Mean-Value Theorem for mutl iterals 400<br/>Exercises 402<br/>Chaqter 15 Multiple Lebesgue Integrals<br/>15.1 Introduction . . .... 405<br/>15.2 Step functions and their integrals ... . 406<br/>15.3 Upper functions and Lebesgue-integrable functions . 406<br/>15.4 Measurable functions and measurable sets in Rn . . 407<br/>15.5 Fubini’s reduction theorem for the double integral of a step function . 409<br/>15.6 Some properties of sets of measure zero ... 411<br/>15.7 Fubini’s reduction theorem for double integrals . 413<br/>15.8 The Tonelli—Hobson test for integra bility . 415<br/>15.9 Coordinate transformations ..... . 416<br/>15.10 The transformation formula for mutl iterals . . 421<br/>15.ll Proof of the transformation formula for linear coordinate transforma-tions 421<br/>15.12 Proof of the transformation formula for the characteristic function of a compact cube 4<br/>15.13 Comlio of the proof of the transformation formula 429<br/>Exercises421<br/>Chapter 16 Cauchy’s Theorem and the Residue Calculus<br/>16.1 Analyticfunctions .. . . . 434<br/>16.2 Paths and curves in the complex plane . .. 435<br/>16.3 Contourintegrals . ... . 436<br/>16.4 The integral along a circular path as a function of the radius . 438<br/>16.5 Cauchy’s integral theorem fora circle ... 439<br/>16.6 Homotopiccurves . ... . . 439<br/>16.7 Invariance of contour integrals under homotopy 442<br/>16.8 General form of Cauchy’s integral theorem . . 443<br/>16.9 Cauchy’si ntegralformula . . .. . 443<br/>16.10 The winding number of a circuit with respect to a point . 444<br/>16.11 The unboundedness of the set of points with winding number zero 446<br/>16.12 Analytic functions defined by contour integrals 447<br/>16.13 Power-series expansions for analytic functions 449<br/>16.14 Cauchy’s inequalities. Liouville’s theorem 450<br/>16.15 Isolation of the zeros of an analytic function 451<br/>16.16 The identity theorem for analytic functions 452<br/>16.17 The maximum and minimum modulus of an analytic function 453<br/>16.18 The open mapping theorem 454<br/>16.19 Laurent expansions for functions analytic in an annulus 455<br/>16.20 Isolated singularities 457<br/>16.21 The residue of a function anislated singular point . 459<br/>16.22 The Cauchy residue theorem 460<br/>16. Counting zeros and poles in a region . 46t<br/>16.24 Evaluation of real-valued integrals by means of residues 462<br/>16.25 Evaluation of Gauss’s sum by residue calculus 464<br/>16.26 Application of the residue theorem to the inversion formula for Laplace transforms 468<br/>16.27 Conformal mappings 470<br/>Exercises 472<br/>Index of Special Symbols 481<br/>Index . 485

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，于1948年在加州大学伯克利分校获得数学博士。