Mathematical Analysis, Second Edition

Cover......Page 1
Copyright page......Page 2
Preface......Page 3
Local interdependence of chapters......Page 4
CONTENTS......Page 5
1.2 The field axioms......Page 15
1.3 The order axioms......Page 16
1.5 Intervals......Page 17
1.7 The unique factorization theorem for integers......Page 18
1.8 Rational numbers......Page 20
1.9 Irrational numbers......Page 21
1.10 Upper bounds, maximum element, least upper bound (supremum)......Page 22
1.12 Some properties of the supremum......Page 23
1.14 The Archimedean property of the real-number system......Page 24
1.16 Finite decimal approximations to real numbers......Page 25
1.18 Absolute values and the triangle inequality......Page 26
1.19 The Cauchy-Schwarz inequality......Page 27
1.20 Plus and minus infinity and the extended real number system $\\mathbb{R}^\\ast$......Page 28
1.21 Complex numbers......Page 29
1.22 Geometric representation of complex numbers......Page 31
1.24 Absolute value of a complex number......Page 32
1.26 Complex exponentials......Page 33
1.28 The argument of a complex number......Page 34
1.29 Integral powers and roots of complex numbers......Page 35
1.30 Complex logarithms......Page 36
1.31 Complex powers......Page 37
1.33 Infinity and the extended complex plane $\\mathbb{C}^\\ast$......Page 38
Exercises......Page 39
2.2 Notations......Page 46
2.4 Cartesian product of two sets......Page 47
2.5 Relations and functions......Page 48
2.6 Further terminology concerning functions......Page 49
2.7 One-to-one functions and inverses......Page 50
2.9 Sequences......Page 51
2.11 Finite and infinite sets......Page 52
2.13 Uncountability of the real-number system......Page 53
2.14 Set algebra......Page 54
2.15 Countable collections of countable sets......Page 56
Exercises......Page 57
3.2 Euclidean space $\\mathbb{R}^n$......Page 61
3.3 Open balls and open sets in $\\mathbb{R}^n$......Page 63
3.4 The structure of open sets in $\\mathbb{R}^1$......Page 64
3.6 Adherent points. Accumulation points......Page 66
3.7 Closed sets and adherent points......Page 67
3.8 The Bolzano-Weierstrass theorem......Page 68
3.10 The Lindelöf covering theorem......Page 70
3.11 The Heine-Borel covering theorem......Page 72
3.12 Compactness in $\\mathbb{R}^n$......Page 73
3.13 Metric spaces......Page 74
3.14 Point set topology in metric spaces......Page 75
3.15 Compact subsets of a metric space......Page 77
3.16 Boundary of a set......Page 78
Exercises......Page 79
4.2 Convergent sequences in a metric space......Page 84
4.3 Cauchy sequences......Page 86
4.5 Limit of a function......Page 88
4.6 Limits of complex-valued functions......Page 90
4.7 Limits of vector-valued functions......Page 91
4.8 Continuous functions......Page 92
4.9 Continuity of composite functions......Page 93
4.11 Examples of continuous functions......Page 94
4.12 Continuity and inverse images of open or closed sets......Page 95
4.13 Functions continuous on compact sets......Page 96
4.15 Bolzano\'s theorem......Page 98
4.16 Connectedness......Page 100
4.17 Components of a metric space......Page 101
4.18 Arcwise connectedness......Page 102
4.19 Uniform continuity......Page 104
4.20 Uniform continuity and compact sets......Page 105
4.22 Discontinuities of real-valued functions......Page 106
4.23 Monotonie functions......Page 108
Exercises......Page 109
5.2 Definition of derivative......Page 118
5.3 Derivatives and continuity......Page 119
5.5 The chain rule......Page 120
5.6 One-sided derivatives and infinite derivatives......Page 121
5.7 Functions with nonzero derivative......Page 122
5.8 Zero derivatives and local extrema......Page 123
5.10 The Mean-Value Theorem for derivatives......Page 124
5.11 Intermediate-value theorem for derivatives......Page 125
5.12 Taylor\'s formula with remainder......Page 127
5.13 Derivatives of vector-valued functions......Page 128
5.14 Partial derivatives......Page 129
5.15 Differentiation of functions of a complex variable......Page 130
5.16 The Cauchy-Riemann equations......Page 132
Exercises......Page 135
6.2 Properties of monotonie functions......Page 141
6.3 Functions of bounded variation......Page 142
6.4 Total variation......Page 143
6.5 Additive property of total variation......Page 144
6.6 Total variation on $[a,x]$ as a function of $x$......Page 145
6.8 Continuous functions of bounded variation......Page 146
6.9 Curves and paths......Page 147
6.10 Rectifiable paths and arc length......Page 148
6.11 Additive and continuity properties of arc length......Page 149
6.12 Equivalence of paths. Change of parameter......Page 150
Exercises......Page 151
7.1 Introduction......Page 154
7.3 The definition of the Riemann-Stieltjes integral......Page 155
7.4 Linear properties......Page 156
7.6 Change of variable in a Riemann-Stieltjes integral......Page 158
7.7 Reduction to a Riemann integral......Page 159
7.8 Step functions as integrators......Page 161
7.9 Reduction of a Riemann-Stieltjes integral to a finite sum......Page 162
7.10 Euler\'s summation formula......Page 163
7.11 Monotonically increasing integrators. Upper and lower integrals......Page 164
7.13 Riemann\'s condition......Page 167
7.14 Comparison theorems......Page 169
7.15 Integrators of bounded variation......Page 170
7.16 Sufficient conditions for existence of Riemann-Stieltjes integrals......Page 173
7.18 Mean Value Theorems for Riemann-Stieltjes integrals......Page 174
7.19 The integral as a function of the interval......Page 175
7.20 Second fundamental theorem of integral calculus......Page 176
7.21 Change of variable in a Riemann integral......Page 177
7.22 Second Mean-Value Theorem for Riemann integrals......Page 179
7.23 Riemann-Stieltjes integrals depending on a parameter......Page 180
7.25 Interchanging the order of integration......Page 181
7.26 Lebesgue\'s criterion for existence of Riemann integrals......Page 183
7.27 Complex-valued Riemann-Stieltjes integrals......Page 187
Exercises......Page 188
8.2 Convergent and divergent sequences of complex numbers......Page 197
8.3 Limit superior and limit inferior of a real-valued sequence......Page 198
8.5 Infinite series......Page 199
8.6 Inserting and removing parentheses......Page 201
8.7 Alternating series......Page 202
8.9 Real and imaginary parts of a complex series......Page 203
8.11 The geometric series......Page 204
8.12 The integral test......Page 205
8.13 The big oh and little oh notation......Page 206
8.15 Dirichlet\'s test and Abel\'s test......Page 207
8.16 Partial sums of the geometric series $\\Sigma z^n$ on the unit circle $|z| = 1$......Page 209
8.17 Rearrangements of series......Page 210
8.19 Subseries......Page 211
8.20 Double sequences......Page 213
8.21 Double series......Page 214
8.22 Rearrangement theorem for double series......Page 215
8.23 A sufficient condition for equality of iterated series......Page 216
8.24 Multiplication of series......Page 217
8.25 Cesàro summability......Page 219
8.26 Infinite products......Page 220
8.27 Euler\'s product for the Riemann zeta function......Page 223
Exercises......Page 224
9.1 Pointwise convergence of sequences of functions......Page 232
9.2 Examples of sequences of real-valued functions......Page 233
9.3 Definition of uniform convergence......Page 234
9.4 Uniform convergence and continuity......Page 235
9.5 The Cauchy condition for uniform convergence......Page 236
9.6 Uniform convergence of infinite series of functions......Page 237
9.7 A space-filling curve......Page 238
9.8 Uniform convergence and Riemann-Stieltjes integration......Page 239
9.9 Nonuniformly convergent sequences that can be integrated term by term......Page 240
9.10 Uniform convergence and differentiation......Page 242
9.11 Sufficient conditions for uniform convergence of a series......Page 244
9.12 Uniform convergence and double sequences......Page 245
9.13 Mean convergence......Page 246
9.14 Power series......Page 248
9.15 Multiplication of power series......Page 251
9.16 The substitution theorem......Page 252
9.17 Reciprocal of a power series......Page 253
9.18 Real power series......Page 254
9.19 The Taylor\'s series generated by a function......Page 255
9.20 Bernstein\'s theorem......Page 256
9.22 Abel\'s limit theorem......Page 258
9.23 Tauber\'s theorem......Page 260
Exercises......Page 261
10.1 Introduction......Page 266
10.2 The integral of a step function......Page 267
10.3 Monotonie sequences of step functions......Page 268
10.4 Upper functions and their integrals......Page 270
10.5 Riemann-integrable functions as examples of upper functions......Page 273
10.6 The class of Lebesgue-integrable functions on a general interval......Page 274
10.7 Basic properties of the Lebesgue integral......Page 275
10.8 Lebesgue integration and sets of measure zero......Page 278
10.9 The Levi monotone convergence theorems......Page 279
10.10 The Lebesgue dominated convergence theorem......Page 284
10.11 Applications of Lebesgue\'s dominated convergence theorem......Page 286
10.12 Lebesgue integrals on unbounded intervals as limits of integrals on bounded intervals......Page 288
10.13 Improper Riemann integrals......Page 290
10.14 Measurable functions......Page 293
10.15 Continuity of functions defined by Lebesgue integrals......Page 295
10.16 Differentiation under the integral sign......Page 297
10.17 Interchanging the order of integration......Page 301
10.18 Measurable sets on the real line......Page 303
10.19 The Lebesgue integral over arbitrary subsets of $\\mathbb{R}$......Page 305
10.20 Lebesgue integrals of complex-valued functions......Page 306
10.21 Inner products and norms......Page 307
10.22 The set $L^2(I)$ of square-integrable functions......Page 308
10.24 A convergence theorem for series of functions in $L^2(I)$......Page 309
10.25 The Riesz-Fischer theorem......Page 311
Exercises......Page 312
11.2 Orthogonal systems of functions......Page 320
11.3 The theorem on best approximation......Page 321
11.5 Properties of the Fourier coefficients......Page 323
11.6 The Riesz-Fischer theorem......Page 325
11.7 The convergence and representation problems for trigonometric series......Page 326
11.8 The Riemann-Lebesgue lemma......Page 327
11.9 The Dirichlet integrals......Page 328
11.10 An integral representation for the partial sums of a Fourier series......Page 331
11.11 Riemann\'s localization theorem......Page 332
11.13 Cesàro summability of Fourier series......Page 333
11.14 Consequences of Fejér\'s theorem......Page 335
11.16 Other forms of Fourier series......Page 336
11.17 The Fourier integral theorem......Page 337
11.18 The exponential form of the Fourier integral theorem......Page 339
11.19 Integral transforms......Page 340
11.20 Convolutions......Page 341
11.21 The convolution theorem for Fourier transforms......Page 343
11.22 The Poisson summation formula......Page 346
Exercises......Page 349
12.2 The directional derivative......Page 358
12.3 Directional derivatives and continuity......Page 359
12.4 The total derivative......Page 360
12.5 The total derivative expressed in terms of partial derivatives......Page 361
12.6 An application to complex-valued functions......Page 362
12.7 The matrix of a linear function......Page 363
12.8 The Jacobian matrix......Page 365
12.9 The chain rule......Page 366
12.10 Matrix form of the chain rule......Page 367
12.11 The Mean-Value Theorem for differentiable functions......Page 369
12.12 A sufficient condition for differentiability......Page 371
12.13 A sufficient condition for equality of mixed partial derivatives......Page 372
12.14 Taylor\'s formula for functions from $\\mathbb{R}^n$ to $\\mathbb{R}^1$......Page 375
Exercises......Page 376
13.1 Introduction......Page 381
13.2 Functions with nonzero Jacobian determinant......Page 382
13.3 The inverse function theorem......Page 386
13.4 The implicit function theorem......Page 387
13.5 Extrema of real-valued functions of one variable......Page 389
13.6 Extrema of real-valued functions of several variables......Page 390
13.7 Extremum problems with side conditions......Page 394
Exercises......Page 398
14.2 The measure of a bounded interval in $\\mathbb{R}^n$......Page 402
14.3 The Riemann integral of a bounded function defined on a compact interval in $\\mathbb{R}^n$......Page 403
14.5 Evaluation of a multiple integral by iterated integration......Page 405
14.6 Jordan-measurable sets in $\\mathbb{R}^n$......Page 410
14.7 Multiple integration over Jordan-measurable sets......Page 411
14.8 Jordan content expressed as a Riemann integral......Page 412
14.9 Additive property of the Riemann integral......Page 413
14.10 Mean-Value Theorem for multiple integrals......Page 414
Exercises......Page 416
15.1 Introduction......Page 419
15.3 Upper functions and Lebesgue-integrable functions......Page 420
15.4 Measurable functions and measurable sets in $\\mathbb{R}^n$......Page 421
15.5 Fubini\'s reduction theorem for the double integral of a step function......Page 423
15.6 Some properties of sets of measure zero......Page 425
15.7 Fubini\'s reduction theorem for double integrals......Page 427
15.8 The Tonelli-Hobson test for integrability......Page 429
15.9 Coordinate transformations......Page 430
15.11 Proof of the transformation formula for linear coordinate transformations......Page 435
15.12 Proof of the transformation formula for the characteristic function of a compact cube......Page 437
15.13 Completion of the proof of the transformation formula......Page 443
Exercises......Page 444
16.1 Analytic functions......Page 448
16.2 Paths and curves in the complex plane......Page 449
16.3 Contour integrals......Page 450
16.4 The integral along a circular path as a function of the radius......Page 452
16.6 Homotopic curves......Page 453
16.7 Invariance of contour integrals under homotopy......Page 456
16.9 Cauchy\'s integral formula......Page 457
16.10 The winding number of a circuit with respect to a point......Page 458
16.11 The unboundedness of the set of points with winding number zero......Page 460
16.12 Analytic functions defined by contour integrals......Page 461
16.13 Power-series expansions for analytic functions......Page 463
16.14 Cauchy\'s inequalities. Liouville\'s theorem......Page 464
16.15 Isolation of the zeros of an analytic function......Page 465
16.16 The identity theorem for analytic functions......Page 466
16.17 The maximum and minimum modulus of an analytic function......Page 467
16.18 The open mapping theorem......Page 468
16.19 Laurent expansions for functions analytic in an annulus......Page 469
16.20 Isolated singularities......Page 471
16.21 The residue of a function at an isolated singular point......Page 473
16.22 The Cauchy residue theorem......Page 474
16.23 Counting zeros and poles in a region......Page 475
16.24 Evaluation of real-valued integrals by means of residues......Page 476
16.25 Evaluation of Gauss\'s sum by residue calculus......Page 478
16.26 Application of the residue theorem to the inversion formula for Laplace transforms......Page 482
16.27 Conformai mappings......Page 484
Exercises......Page 486
Index of Special Symbols......Page 495
Index......Page 499