数学分析英文版第2版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Chapter 1 The Real and Complex Number Systems1

1.1 Introduction1

1.2 The field axioms1

1.3 The order axioms2

1.4 Geometric representation of real numbers3

1.5 Intervals3

1.6 Integers4

1.7 The unique factorization theorem for integers4

1.8 Rational numbers6

1.9 Irrational numbers7

1.10 Upper bounds,maximum element,least upper bound(supremum)8

1.11 The completeness axiom9

1.12 Some properties of the supremum9

1.13 Properties of the integers deduced from the completeness axiom10

1.14 The Archimedean property of the real-number system10

1.15 Rational numbers with finite decimal representation11

1.16 Finite decimal approximations to real numbers11

1.17 Infinite decimal representation of real numbers12

1.18 Absolute values and the triangle inequality12

1.19 The Cauchy-Schwarz inequality13

1.20 Plus and minus infinity and the extended real number svstem R14

1.21 Complex numbers15

1.22 Geometric representation of complex numbers17

1.23 The imaginary unit18

1.24 Absolute value of a complex number18

1.25 Impossibility of ordering the complex numbers19

1.26 Complex exponentials19

1.27 Further properties of complex exponentials20

1.28 The argument of a complex number20

1.29 Integral powers and roots of complex numbers21

1.30 Complex logarithms22

1.31 Complex powers23

1.32 Complex sines and cosines24

1.33 Infinity and the extended complex plane C24

Exercises25

Chapter 2 Some Basic Notions of Set Theory32

2.1 Introduction32

2.2 Notations32

2.3 Ordered pairs33

2.4 Cartesian product of two sets33

2.5 Relations and functions34

2.6 Further terminology concerning functions35

2.7 One-to-one functions and inverses36

2.8 Composite functions37

2.9 Sequences37

2.10 Similar(equinumerous)sets38

2.11 Finite and infinite sets38

2.12 Countable and uncountable sets39

2.13 Uncountability of the real-number system39

2.14 Set algebra40

2.15 Countable collections of countable sets42

Exercises43

Chapter 3 Elements of Point Set Topology47

3.1 Introduction47

3.2 Euclidean space Rn47

3.3 Open balls and open sets in Rn49

3.4 The structure of open sets in R150

3.5 Closed sets52

3.6 Adherent points.Accumulation points52

3.7 Closed sets and adherent points53

3.8 The Bolzano-Weierstrass theorem54

3.9 The Cantor intersection theorem56

3.10 The Lindel？f covering theorem56

3.1l The Heine-Borel covering theorem58

3.12 Compactness in Rn59

3.13 Metric spaces60

3.14 Point set topology in metric spaces61

3.15 Compact subsets of a metric space63

3.16 Boundary of a set64

Exercises65

Chapter 4 Limits and Continuity70

4.1 Introduction70

4.2 Convergent sequences in a metric space70

4.3 Cauchy sequences72

4.4 Complete metric spaces74

4.5 Limit of a function74

4.6 Limits of complex-valued functions76

4.7 Limits of vector-valued functions77

4.8 Continuous functions78

4.9 Continuity of composite functions79

4.10 Continuous complex-valued and vector-valued functions80

4.11 Examples of continuous functions80

4.12 Continuity and inverse images ofopen or closed sets81

4.13 Functions continuous on compact sets82

4.14 Topological mappings(homeomorphisms)84

4.15 Bolzano's theorem84

4.16 Connectedness86

4.17 Components of a metric space87

4.18 Arcwise connectedness88

4.19 Uniform continuity90

4.20 Uniform continuity and compact sets91

4.21 Fixed-point theorem for contractions92

4.22 Discontinuities of real-valued functions92

4.23 Monotonic functions94

Exercises95

Chapter 5 Derivatives104

5.1 Introduction104

5.2 Definition of derivative104

5.3 Derivatives and continuity105

5.4 Algebra of derivatives106

5.5 The chain rule106

5.6 One-sided derivatives and infinite derivatives107

5.7 Functions with nonzero derivative108

5.8 Zero derivatives and local extrema109

5.9 Rolle's theorem110

5.10 The Mean-Value Theorem for derivatives110

5.11 Intermediate-value theorem for derivatives111

5.12 Taylor's formula with remainder113

5.13 Derivatives of vector-valued functions114

5.14 Partial derivatives115

5.15 Differentiation of functions of a complex variable116

5.16 The Cauchy-Riemann equations118

Exercises121

Chapter 6 Functions of Bounded Variation and Rectifiable Curves127

6.1 Introduction127

6.2 Properties of monotonic functions127

6.3 Functions of bounded variation128

6.4 Total variation129

6.5 Additive property of total variation130

6.6 Total variation on[a,x]as a function of x131

6.7 Functions of bounded variation expressed as the difference of increasing functions132

6.8 Continuous functions of bounded variation132

6.9 Curves and paths133

6.10 Rectifiable paths and arc length134

6.11 Additive and continuity properties of arc length135

6.12 Equivalence of paths.Change of parameter136

Exercises137

Chapter 7 The Riemann-Stieltjes Integral140

7.1 Introduction140

7.2 Notation141

7.3 The definition of the Riemann-Stieltjes integral141

7.4 Linear properties142

7.5 Integration by parts144

7.6 Change of variable in a Riemann-Stieltjes integral144

7.7 Reduction to a Riemann integral145

7.8 Step functions as integrators147

7.9 Reduction of a Riemann-Stieltjes integral to a finite sum148

7.10 Euler's summation formula149

7.11 Monotonically increasing integrators.Upper and lower integrals150

7.12 Additive and linearity properties of upper and lower integrals153

7.13 Riemann's condition153

7.14 Comparison theorems155

7.15 Integrators of bounded variation156

7.16 Sufficient conditions for existence of Riemann-Stieltjes integrals159

7.17 Necessary conditions for existence of Riemann-Stieltjes integrals160

7.18 Mean Value Theorems for Riemann-Stieltjes integrals160

7.19 The integral as a function of the interval161

7.20 Second fundamental theorem of integral calculus162

7.21 Change of variable in a Riemann integral163

7.22 Second Mean-Value Theorem for Riemann integrals165

7.23 Riemann-Stieltjes integrals depending on a parameter166

7.24 Differentiation under the integral sign167

7.25 Interchanging the order of integration167

7.26 Lebesgue's criterion for existence of Riemann integrals169

7.27 Complex-valued Riemann-Stieltjes integrals173

Exercises174

Chapter 8 Infinite Series and Infinite Products183

8.1 Introduction183

8.2 Convergent and divergent sequences of complex numbers183

8.3 Limit superior and limit inferior of a real-valued sequence184

8.4 Monotonic sequences of real numbers185

8.5 Infinite series185

8.6 Inserting and removing parentheses187

8.7 Alternating series188

8.8 Absolute and conditional convergence189

8.9 Real and imaginary parts of a complex series189

8.10 Tests for convergence of series with positive terms190

8.11 The geometric series190

8.12 The integral test191

8.13 The big oh and little oh notation192

8.14 The ratio test and the root test193

8.15 Dirichlet's test and Abel's test193

8.16 Partial sums of the geometric series ∑zn on the unit circle |z|=1195

8.17 Rearrangements of series196

8.18 Riemann's theorem on conditionally convergent series197

8.19 Subseries197

8.20 Double sequences199

8.21 Double series200

8.22 Rearrangement theorem for double series201

8.23 A sufficient condition for equality of iterated series202

8.24 Multiplication of series203

8.25 Cesàro summability205

8.26 Infinite products206

8.27 Euler's product for the Riemann zeta function209

Exercises210

Chapter 9 Sequences of Functions218

9.1 Pointwise convergence of sequences of functions218

9.2 Examples of sequences of real-valued functions219

9.3 Definition of uniform convergence220

9.4 Uniform convergence and continuity221

9.5 The Cauchy condition for uniform convergence222

9.6 Uniform convergence of infinite series of functions223

9.7 A space-filling curve224

9.8 Uniform convergence and Riemann-Stieltjes integration225

9.9 Nonuniformly convergent sequences that can be integrated term by term226

9.10 Uniform convergence and differentiation228

9.11 Sufficient conditions for uniform convergence of a series230

9.12 Uniform convergence and double sequences231

9.13 Mean convergence232

9.14 Power series234

9.15 Multiplication of power series237

9.16 The substitution theorem238

9.17 Reciprocal of a power series239

9.18 Real power series240

9.19 The Taylor's series generated by a function241

9.20 Bernstein's theorem242

9.21 The binomial series244

9.22 Abel's limit theorem244

9.23 Tauber's theorem246

Exercises247

Chapter 10 The Lebesgue Integral252

10.1 Introduction252

10.2 The integral of a step function253

10.3 Monotonic sequences of step functions254

10.4 Upper functions and their integrals256

10.5 Riemann-integrable functions as examples of upper functions259

10.6 The class of Lebesgue-integrable functions on a general interval260

10.7 Basic properties of the Lebesgue integral261

10.8 Lebesgue integration and sets of measure zero264

10.9 The Levi monotone convergence theorems265

10.10 The Lebesgue dominated convergence theorem270

10.11 Applications of Lebesgue's dominated convergence theorem272

10.12 Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals274

10.13 Improper Riemann integrals276

10.14 Measurable functions279

10.15 Continuity of functions defined by Lebesgue integrals281

10.16 Differentiation under the integral sign283

10.17 Interchanging the order of integration287

10.18 Measurable sets on the real line289

10.19 The Lebesgue integral over arbitrary subsets of R291

10.20 Lebesgue integrals of complex-valued functions292

10.21 Inner products and norms293

10.22 The set L2(I)of square-integrable functions294

10.23 The set L2(I)as a semimetric space295

10.24 A convergence theorem for series of functions in L2(I)295

10.25 The Riesz-Fischer theorem297

Exercises298

Chapter 11 Fourier Series and Fourier Integrals306

11.1 Introduction306

11.2 Orthogonal systems of functions306

11.3 The theorem on best approximation307

11.4 The Fourier series of a function relative to an orthonormal system309

11.5 Properties of the Fourier coefficients309

11.6 The Riesz-Fischer theorem311

11.7 Theconvergence and representation problems for trigonometric series312

11.8 The Riemann-Lebesgue lemma313

11.9 The Dirichlet integrals314

11.10 An integral representation for the partial sums of a Fourier series317

11.11 Riemann's localization theorem318

11.12 Sufficient conditions for convergence of a Fourier series at a particular point319

11.13 Cesàro summability of Fourier series319

11.14 Consequences of Fejér's theorem321

11.15 The Weierstrass approximation theorem322

11.16 Other forms of Fourier series322

11.17 The Fourier integral theorem323

11.18 The exponential form of the Fourier integral theorem325

11.19 Integral transforms326

11.20 Convolutions327

11.21 The convolution theorem for Fourier transforms329

11.22 The Poisson summation formula332

Exercises335

Chapter 12 Multivariable Differential Calculus344

12.1 Introduction344

12.2 The directional derivative344

12.3 Directional derivatives and continuity345

12.4 The total derivative346

12.5 The total derivative expressed in terms of partial derivatives347

12.6 An application to complex-valued functions348

12.7 The matrix of a linear function349

12.8 The Jacobian matrix351

12.9 The chain rule352

12.10 Matrix form of the chain rule353

12.11 The Mean-Value Theorem for differentiable functions355

12.12 A sufficient condition for differentiability357

12.13 A sufficient condition for equality of mixed partial derivatives358

12.14 Taylor's formula for functions from Rn to R1361

Exercises362

Chapter 13 Implicit Functions and Extremum Problems367

13.1 Introduction367

13.2 Functions with nonzero Jacobian determinant368

13.3 The inverse function theorem372

13.4 The implicit function theorem373

13.5 Extrema of real-valued functions of one variable375

13.6 Extrema of real-valued functions of several variables376

13.7 Extremum problems with side conditions380

Exercises384

Chapter 14 Multiple Riemann Integrals388

14.1 Introduction388

14.2 The measure of a bounded interval in Rn388

14.3 The Riemann integral of a bounded function defined on a compact interval in Rn389

14.4 Sets of measure zero and Lebesgue's criterion for existence of a multiple Riemann integral391

14.5 Evaluation of a multiple integral by iterated integration391

14.6 Jordan-measurable sets in Rn396

14.7 Multiple integration over Jordan-measurable sets397

14.8 Jordan content expressed as a Riemann integral398

14.9 Additive property of the Riemann integral399

14.10 Mean-Value Theorem for multiple integrals400

Exercises402

Chapter 15 Multiple Lebesgue Integrals405

15.1 Introduction405

15.2 Step functions and their integrals406

15.3 Upper functions and Lebesgue-integrable functions406

15.4 Measurable functions and measurable sets in Rn407

15.5 Fubini's reduction theorem for the double integral of a step function409

15.6 Some properties of sets of measure zero411

15.7 Fubini's reduction theorem for double integrals413

15.8 The Tonelli-Hobson test for integrability415

15.9 Coordinate transformations416

15.10 The transformation formula for multiple integrals421

15.11 Proof of the transformation formula for linear coordinate transformations421

15.12 Proof of the transformation formula for the characteristic function of a compact cube423

15.13 Completion of the proof of the transformation formula429

Exercises430

Chapter 16 Cauchy's Theorem and the Residue Calculus434

16.1 Analytic functions434

16.2 Paths and curves in the complex plane435

16.3 Contour integrals436

16.4 The integral along a circular path as a function of the radius438

16.5 Cauchy's integral theorem for a circle439

16.6 Homotopic curves439

16.7 Invariance of contour integrals under homotopy442

16.8 General form of Cauchy's integral theorem443

16.9 Cauchy's integral formula443

16.10 The winding number of a circuit with respect to a point444

16.11 The unboundedness of the set of points with winding number zero446

16.12 Analytic functions defined by contour integrals447

16.13 Power-series expansions for analytic functions449

16.14 Cauchy's inequalities.Liouville's theorem450

16.15 Isolation of the zeros of an analytic function451

16.16 The identity theorem for analytic functions452

16.17 The maximum and minimum modulus of an analytic function453

16.18 The open mapping theorem454

16.19 Laurent expansions for functions analytic in an annulus455

16.20 Isolated singularities457

16.21 The residue of a function at an isolated singular point459

16.22 The Cauchy residue theorem460

16.23 Counting zeros and poles in a region461

16.24 Evaluation of real-valued integrals by means of residues462

16.25 Evaluation of Gauss's sum by residue calculus464

16.26 Application ofthe residue theorem to the inversion formula for Laplace transforms468

16.27 Conformal mappings470

Exercises472

Index of Special Symbols481

Index485