Chapter 1 The Real and Complex Number Systems 1
1.1 Introduction 1
1.2 The field axioms 1
1.3 The order axioms 2
1.4 Geometric representation of real numbers 3
1.5 Intervals 3
1.6 Integers 4
1.7 The unique factorization theorem for integers 4
1.8 Rational numbers 6
1.9 Irrational numbers 7
1.10 Upper bounds,maximum element,least upper bound(supremum) 8
1.11 The completeness axiom 9
1.12 Some properties of the supremum 9
1.13 Properties of the integers deduced from the completeness axiom 10
1.14 The Archimedean property of the real-number system 10
1.15 Rational numbers with finite decimal representation 11
1.16 Finite decimal approximations to real numbers 11
1.17 Infinite decimal representation of real numbers 12
1.18 Absolute values and the triangle inequality 12
1.19 The Cauchy-Schwarz inequality 13
1.20 Plus and minus infinity and the extended real number svstem R 14
1.21 Complex numbers 15
1.22 Geometric representation of complex numbers 17
1.23 The imaginary unit 18
1.24 Absolute value of a complex number 18
1.25 Impossibility of ordering the complex numbers 19
1.26 Complex exponentials 19
1.27 Further properties of complex exponentials 20
1.28 The argument of a complex number 20
1.29 Integral powers and roots of complex numbers 21
1.30 Complex logarithms 22
1.31 Complex powers 23
1.32 Complex sines and cosines 24
1.33 Infinity and the extended complex plane C 24
Exercises 25
Chapter 2 Some Basic Notions of Set Theory 32
2.1 Introduction 32
2.2 Notations 32
2.3 Ordered pairs 33
2.4 Cartesian product of two sets 33
2.5 Relations and functions 34
2.6 Further terminology concerning functions 35
2.7 One-to-one functions and inverses 36
2.8 Composite functions 37
2.9 Sequences 37
2.10 Similar(equinumerous)sets 38
2.11 Finite and infinite sets 38
2.12 Countable and uncountable sets 39
2.13 Uncountability of the real-number system 39
2.14 Set algebra 40
2.15 Countable collections of countable sets 42
Exercises 43
Chapter 3 Elements of Point Set Topology 47
3.1 Introduction 47
3.2 Euclidean space Rn 47
3.3 Open balls and open sets in Rn 49
3.4 The structure of open sets in R1 50
3.5 Closed sets 52
3.6 Adherent points.Accumulation points 52
3.7 Closed sets and adherent points 53
3.8 The Bolzano-Weierstrass theorem 54
3.9 The Cantor intersection theorem 56
3.10 The Lindel？f covering theorem 56
3.1l The Heine-Borel covering theorem 58
3.12 Compactness in Rn 59
3.13 Metric spaces 60
3.14 Point set topology in metric spaces 61
3.15 Compact subsets of a metric space 63
3.16 Boundary of a set 64
Exercises 65
Chapter 4 Limits and Continuity 70
4.1 Introduction 70
4.2 Convergent sequences in a metric space 70
4.3 Cauchy sequences 72
4.4 Complete metric spaces 74
4.5 Limit of a function 74
4.6 Limits of complex-valued functions 76
4.7 Limits of vector-valued functions 77
4.8 Continuous functions 78
4.9 Continuity of composite functions 79
4.10 Continuous complex-valued and vector-valued functions 80
4.11 Examples of continuous functions 80
4.12 Continuity and inverse images ofopen or closed sets 81
4.13 Functions continuous on compact sets 82
4.14 Topological mappings(homeomorphisms) 84
4.15 Bolzano's theorem 84
4.16 Connectedness 86
4.17 Components of a metric space 87
4.18 Arcwise connectedness 88
4.19 Uniform continuity 90
4.20 Uniform continuity and compact sets 91
4.21 Fixed-point theorem for contractions 92
4.22 Discontinuities of real-valued functions 92
4.23 Monotonic functions 94
Exercises 95
Chapter 5 Derivatives 104
5.1 Introduction 104
5.2 Definition of derivative 104
5.3 Derivatives and continuity 105
5.4 Algebra of derivatives 106
5.5 The chain rule 106
5.6 One-sided derivatives and infinite derivatives 107
5.7 Functions with nonzero derivative 108
5.8 Zero derivatives and local extrema 109
5.9 Rolle's theorem 110
5.10 The Mean-Value Theorem for derivatives 110
5.11 Intermediate-value theorem for derivatives 111
5.12 Taylor's formula with remainder 113
5.13 Derivatives of vector-valued functions 114
5.14 Partial derivatives 115
5.15 Differentiation of functions of a complex variable 116
5.16 The Cauchy-Riemann equations 118
Exercises 121
Chapter 6 Functions of Bounded Variation and Rectifiable Curves 127
6.1 Introduction 127
6.2 Properties of monotonic functions 127
6.3 Functions of bounded variation 128
6.4 Total variation 129
6.5 Additive property of total variation 130
6.6 Total variation on[a,x]as a function of x 131
6.7 Functions of bounded variation expressed as the difference of increasing functions 132
6.8 Continuous functions of bounded variation 132
6.9 Curves and paths 133
6.10 Rectifiable paths and arc length 134
6.11 Additive and continuity properties of arc length 135
6.12 Equivalence of paths.Change of parameter 136
Exercises 137
Chapter 7 The Riemann-Stieltjes Integral 140
7.1 Introduction 140
7.2 Notation 141
7.3 The definition of the Riemann-Stieltjes integral 141
7.4 Linear properties 142
7.5 Integration by parts 144
7.6 Change of variable in a Riemann-Stieltjes integral 144
7.7 Reduction to a Riemann integral 145
7.8 Step functions as integrators 147
7.9 Reduction of a Riemann-Stieltjes integral to a finite sum 148
7.10 Euler's summation formula 149
7.11 Monotonically increasing integrators.Upper and lower integrals 150
7.12 Additive and linearity properties of upper and lower integrals 153
7.13 Riemann's condition 153
7.14 Comparison theorems 155
7.15 Integrators of bounded variation 156
7.16 Sufficient conditions for existence of Riemann-Stieltjes integrals 159
7.17 Necessary conditions for existence of Riemann-Stieltjes integrals 160
7.18 Mean Value Theorems for Riemann-Stieltjes integrals 160
7.19 The integral as a function of the interval 161
7.20 Second fundamental theorem of integral calculus 162
7.21 Change of variable in a Riemann integral 163
7.22 Second Mean-Value Theorem for Riemann integrals 165
7.23 Riemann-Stieltjes integrals depending on a parameter 166
7.24 Differentiation under the integral sign 167
7.25 Interchanging the order of integration 167
7.26 Lebesgue's criterion for existence of Riemann integrals 169
7.27 Complex-valued Riemann-Stieltjes integrals 173
Exercises 174
Chapter 8 Infinite Series and Infinite Products 183
8.1 Introduction 183
8.2 Convergent and divergent sequences of complex numbers 183
8.3 Limit superior and limit inferior of a real-valued sequence 184
8.4 Monotonic sequences of real numbers 185
8.5 Infinite series 185
8.6 Inserting and removing parentheses 187
8.7 Alternating series 188
8.8 Absolute and conditional convergence 189
8.9 Real and imaginary parts of a complex series 189
8.10 Tests for convergence of series with positive terms 190
8.11 The geometric series 190
8.12 The integral test 191
8.13 The big oh and little oh notation 192
8.14 The ratio test and the root test 193
8.15 Dirichlet's test and Abel's test 193
8.16 Partial sums of the geometric series ∑zn on the unit circle |z|=1 195
8.17 Rearrangements of series 196
8.18 Riemann's theorem on conditionally convergent series 197
8.19 Subseries 197
8.20 Double sequences 199
8.21 Double series 200
8.22 Rearrangement theorem for double series 201
8.23 A sufficient condition for equality of iterated series 202
8.24 Multiplication of series 203
8.25 Cesàro summability 205
8.26 Infinite products 206
8.27 Euler's product for the Riemann zeta function 209
Exercises 210
Chapter 9 Sequences of Functions 218
9.1 Pointwise convergence of sequences of functions 218
9.2 Examples of sequences of real-valued functions 219
9.3 Definition of uniform convergence 220
9.4 Uniform convergence and continuity 221
9.5 The Cauchy condition for uniform convergence 222
9.6 Uniform convergence of infinite series of functions 223
9.7 A space-filling curve 224
9.8 Uniform convergence and Riemann-Stieltjes integration 225
9.9 Nonuniformly convergent sequences that can be integrated term by term 226
9.10 Uniform convergence and differentiation 228
9.11 Sufficient conditions for uniform convergence of a series 230
9.12 Uniform convergence and double sequences 231
9.13 Mean convergence 232
9.14 Power series 234
9.15 Multiplication of power series 237
9.16 The substitution theorem 238
9.17 Reciprocal of a power series 239
9.18 Real power series 240
9.19 The Taylor's series generated by a function 241
9.20 Bernstein's theorem 242
9.21 The binomial series 244
9.22 Abel's limit theorem 244
9.23 Tauber's theorem 246
Exercises 247
Chapter 10 The Lebesgue Integral 252
10.1 Introduction 252
10.2 The integral of a step function 253
10.3 Monotonic sequences of step functions 254
10.4 Upper functions and their integrals 256
10.5 Riemann-integrable functions as examples of upper functions 259
10.6 The class of Lebesgue-integrable functions on a general interval 260
10.7 Basic properties of the Lebesgue integral 261
10.8 Lebesgue integration and sets of measure zero 264
10.9 The Levi monotone convergence theorems 265
10.10 The Lebesgue dominated convergence theorem 270
10.11 Applications of Lebesgue's dominated convergence theorem 272
10.12 Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals 274
10.13 Improper Riemann integrals 276
10.14 Measurable functions 279
10.15 Continuity of functions defined by Lebesgue integrals 281
10.16 Differentiation under the integral sign 283
10.17 Interchanging the order of integration 287
10.18 Measurable sets on the real line 289
10.19 The Lebesgue integral over arbitrary subsets of R 291
10.20 Lebesgue integrals of complex-valued functions 292
10.21 Inner products and norms 293
10.22 The set L2(I)of square-integrable functions 294
10.23 The set L2(I)as a semimetric space 295
10.24 A convergence theorem for series of functions in L2(I) 295
10.25 The Riesz-Fischer theorem 297
Exercises 298
Chapter 11 Fourier Series and Fourier Integrals 306
11.1 Introduction 306
11.2 Orthogonal systems of functions 306
11.3 The theorem on best approximation 307
11.4 The Fourier series of a function relative to an orthonormal system 309
11.5 Properties of the Fourier coefficients 309
11.6 The Riesz-Fischer theorem 311
11.7 Theconvergence and representation problems for trigonometric series 312
11.8 The Riemann-Lebesgue lemma 313
11.9 The Dirichlet integrals 314
11.10 An integral representation for the partial sums of a Fourier series 317
11.11 Riemann's localization theorem 318
11.12 Sufficient conditions for convergence of a Fourier series at a particular point 319
11.13 Cesàro summability of Fourier series 319
11.14 Consequences of Fejér's theorem 321
11.15 The Weierstrass approximation theorem 322
11.16 Other forms of Fourier series 322
11.17 The Fourier integral theorem 323
11.18 The exponential form of the Fourier integral theorem 325
11.19 Integral transforms 326
11.20 Convolutions 327
11.21 The convolution theorem for Fourier transforms 329
11.22 The Poisson summation formula 332
Exercises 335
Chapter 12 Multivariable Differential Calculus 344
12.1 Introduction 344
12.2 The directional derivative 344
12.3 Directional derivatives and continuity 345
12.4 The total derivative 346
12.5 The total derivative expressed in terms of partial derivatives 347
12.6 An application to complex-valued functions 348
12.7 The matrix of a linear function 349
12.8 The Jacobian matrix 351
12.9 The chain rule 352
12.10 Matrix form of the chain rule 353
12.11 The Mean-Value Theorem for differentiable functions 355
12.12 A sufficient condition for differentiability 357
12.13 A sufficient condition for equality of mixed partial derivatives 358
12.14 Taylor's formula for functions from Rn to R1 361
Exercises 362
Chapter 13 Implicit Functions and Extremum Problems 367
13.1 Introduction 367
13.2 Functions with nonzero Jacobian determinant 368
13.3 The inverse function theorem 372
13.4 The implicit function theorem 373
13.5 Extrema of real-valued functions of one variable 375
13.6 Extrema of real-valued functions of several variables 376
13.7 Extremum problems with side conditions 380
Exercises 384
Chapter 14 Multiple Riemann Integrals 388
14.1 Introduction 388
14.2 The measure of a bounded interval in Rn 388
14.3 The Riemann integral of a bounded function defined on a compact interval in Rn 389
14.4 Sets of measure zero and Lebesgue's criterion for existence of a multiple Riemann integral 391
14.5 Evaluation of a multiple integral by iterated integration 391
14.6 Jordan-measurable sets in Rn 396
14.7 Multiple integration over Jordan-measurable sets 397
14.8 Jordan content expressed as a Riemann integral 398
14.9 Additive property of the Riemann integral 399
14.10 Mean-Value Theorem for multiple integrals 400
Exercises 402
Chapter 15 Multiple Lebesgue Integrals 405
15.1 Introduction 405
15.2 Step functions and their integrals 406
15.3 Upper functions and Lebesgue-integrable functions 406
15.4 Measurable functions and measurable sets in Rn 407
15.5 Fubini's reduction theorem for the double integral of a step function 409
15.6 Some properties of sets of measure zero 411
15.7 Fubini's reduction theorem for double integrals 413
15.8 The Tonelli-Hobson test for integrability 415
15.9 Coordinate transformations 416
15.10 The transformation formula for multiple integrals 421
15.11 Proof of the transformation formula for linear coordinate transformations 421
15.12 Proof of the transformation formula for the characteristic function of a compact cube 423
15.13 Completion of the proof of the transformation formula 429
Exercises 430
Chapter 16 Cauchy's Theorem and the Residue Calculus 434
16.1 Analytic functions 434
16.2 Paths and curves in the complex plane 435
16.3 Contour integrals 436
16.4 The integral along a circular path as a function of the radius 438
16.5 Cauchy's integral theorem for a circle 439
16.6 Homotopic curves 439
16.7 Invariance of contour integrals under homotopy 442
16.8 General form of Cauchy's integral theorem 443
16.9 Cauchy's integral formula 443
16.10 The winding number of a circuit with respect to a point 444
16.11 The unboundedness of the set of points with winding number zero 446
16.12 Analytic functions defined by contour integrals 447
16.13 Power-series expansions for analytic functions 449
16.14 Cauchy's inequalities.Liouville's theorem 450
16.15 Isolation of the zeros of an analytic function 451
16.16 The identity theorem for analytic functions 452
16.17 The maximum and minimum modulus of an analytic function 453
16.18 The open mapping theorem 454
16.19 Laurent expansions for functions analytic in an annulus 455
16.20 Isolated singularities 457
16.21 The residue of a function at an isolated singular point 459
16.22 The Cauchy residue theorem 460
16.23 Counting zeros and poles in a region 461
16.24 Evaluation of real-valued integrals by means of residues 462
16.25 Evaluation of Gauss's sum by residue calculus 464
16.26 Application ofthe residue theorem to the inversion formula for Laplace transforms 468
16.27 Conformal mappings 470
Exercises 472
Index of Special Symbols 481
Index 485