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Chapter 1 The Real and Complex Number Systems 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 system 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 2.1 Introductiou 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. 38 2.10 Similar (equinumerous) sets 38 2.11 Finite and infinite sets 39 2.12 Countable and uncountable sets 39 2.13 Uncountability of the real-number system 42 2.14 Set algebra 43 2.15 Countable collections of countable sets Exercises 43 Chapter 3 Elements of Point Set Topology 3.1 Introduction 47 3.2 Euclidean space R't 47 3.3 Open balls and open sets in R 49 >3.4 The structure of open sets in RH 50 3.5 Closed sets . 52 3.6 Adhèrent points. Accumulation points 52 3.7 Closed sets and adhèrent points 53 3.8 The Bolzano—Weierstrass theorem 54 3.9 The Cantor intersection theorem 56 3.10 The Lindelf covering theorem 56 3.11 The Heine—Borel covering theorem 58 3.12 Compactness in R‘ 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 Exercises 65 Chaqter 4 Limits and Continuity 4.1 Introduction 70 4.2 Convergent sequences in a metric space 72 4.3 Cauchy sequences 74 4.4 Complete metric spaces . 74 4.5 Limit of a function 76 4.6 Limits of complex-valued functions 4.7 Limits of vector-valued functions 77 4.8 Continuous functions 78 4.9 Continuity of composite functions. 4.10 Continuous complex-valued and vector-valued functions 79 4.11 Examples of continuous functions 80 4.12 Continuity and inverse images of open or closed sets 80 4.13 Functions continuous on compact sets 81 4.14 Topolo＄ical mappings (homeomorphisms) 82 4.15 Bolzano’s theorem 84 4.16 Connectedness 84 4.17 Components of a metric space . 86 4.18 Arcwise connectedness 87 4.19 Uniform continuity 88 4.20 Uniform continuity and compact sets 90 4.21 Fixed-point theorem for contractions 91 4.22 Discontinuities of real-valued functions 92 4.23 Monotonic functions 94 Exercises 95 Chapter 5 DerJvatives 5.1Introduction 104 5.2 Definition of derivative .104 5.3 Derivatives and continuity 105 5.4 Algebra of derivatives106 5.5 The chain rule 106 5.6 One-sided derivatives and infinite derivatives 106 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 DiPerentiation of functions of a complex variable 116 5.16 The Cauchy Riemann equations 118 Exercises 121 Chapter 6 Functions of Bounded Variation and Reetifiable Curves 6.1 Introduction 127 6.2 PropertleS Of monotonic functions 128 6.3 Functions of bounded variation 129 6.4 Total variation 130 6.5 Additive property of total variation 131 6.6 Total variation on (a, x) as a function of x 132 6.7 Functions of bounded variation expressed as the diPerence of increasing functions . 133 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 136 Chapter 7 The Riemann—Stieltjes Integral 7.1 Introduction 140 7.2 Notation 141 7.3 The definition of the Riemann—Stieltjes integral 141 7.4 Linear properties 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 Dikerentiation 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 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 parenthèses 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 Z z‘ on the unit circle ]z] 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 Rearran＄ement 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 209 8.27 Euler’s product for the Riemann zeta function 209 Exercises 210 Chaqter 9 Sequences of Functions 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 226 9.10 Uniform convergence and diPerentiation 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 246 9.23 Tauber’s theorem 246 Exercises 247 Chapter 10 The Lebesgue Integral 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 internal 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 DiPerentiation 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(/ ) of square-integrable functions . 294 10.23 The set L2(J) 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 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 ihe Fourier coeificients . . . . 309 11.6 The Riesz—Fischer theorem .. .... . 31i >11.7 The convergence and representation problems for trigonometric series 31211.8 The Riemann—Lebesgue lemma 11.9 The Dirichlet integrals 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 320 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 327 11.20 Convolution 327 11.21 The convolution theorem for Fourier transforms 329 11.22 The Poisson summation formula 332 Exercise 335 Chaqter 12 Sufficient conditions for convergence of a Fourier series at a particular point 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 di#erentiable functions 355 12.12 A sufficient condition for diPerentiability 357 12.13 A sufficient condition for equality of mixed partial derivatives 358 12.14 Taylor's formula for functions from R to R' 361 Exercises 362 Chapter 13 Implicit Functions and Extremum Problems 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 Riemam liitegrals 14.1 Introduction 388 14.2 The measure of a bounded internal in R 388 14.3 The Riemann integral of a bounded function defined on a compact 389 interval in R 14.4 Sets of measure zero and kebesgue’s criterion for existence of a multiple Riemann integral391 14.5 Evaluation of a multiple integral by iterated integration 391 14.6 Jordan-measurable sets in R 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 366 14.10 Mean-Value Theorem for multiple integrals 400 Exercises 402 Chaqter 15 Multiple Lebesgue Integrals 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 integra bility . 415 15.9 Coordinate transformations ..... . 416 15.10 The transformation formula for multiple integrals . . 421 15.ll Proof of the transformation formula for linear coordinate transforma-tions 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 Exercises421 Chapter 16 Cauchy’s Theorem and the Residue Calculus 16.1 Analyticfunctions .. . . . 434 16.2 Paths and curves in the complex plane . .. 435 16.3 Contourintegrals . ... . 436 16.4 The integral along a circular path as a function of the radius . 438 16.5 Cauchy’s integral theorem fora circle ... 439 16.6 Homotopiccurves . ... . . 439 16.7 Invariance of contour integrals under homotopy 442 6.8 General form of Cauchy’s integral theorem . . 443 16.9 Cauchy’si ntegralformula . . .. . 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 . 46t 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 of the residue theorem to the inversion formula for Laplace transforms 468 16.27 Conformal mappings 470 Exercises 472 Index of Special Symbols 481 Index . 485
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