Mathematical Analysis

CONTENTS 6
CHAPTER 1 THE REAL AND COMPLEX NUMBER SYSTEMS ... 15
1.1 INTRODUCTION ... 15
1.2 THE FIELD AXIOMS ... 15
1.3 THE ORDER AXIOMS ... 16
1.4 GEOMETRIC REPRESENTATION OF REAL NUMBERS ... 17
1.5 INTERVALS ... 17
1.6 INTEGERS ... 18
1.7 THE UNIQUE FACTORIZATION THEOREM FOR INTEGERS ... 18
1.8 RATIONAL NUMBERS ... 20
1.9 IRRATIONAL NUMBERS ... 21
1.10 UPPER BOUNDS, MAXIMUM ELEMENT, LEAST UPPER BOUND (SUPREMUM) ... 22
1.11 THE COMPLETENESS AXIOM ... 23
1.12 SOME PROPERTIES OF THE SUPREMUM ... 23
1.13 PROPERTIES OF THE INTEGERS DEDUCED FROM THE COMPLETENESS AXIOM ... 24
1.14 THE ARCHIMEDEAN PROPERTY OF THE REAL NUMBER SYSTEM ... 24
1.15 RATIONAL NUMBERS WITH FINITE DECIMAL REPRESENTATION ... 25
1.16 FINITE DECIMAL APPROXIMATIONS TO REAL NUMBERS ... 25
1.17 INFINITE DECIMAL REPRESENTATIONS OF REAL NUMBERS ... 26
1.18 ABSOLUTE VALUES AND THE TRIANGLE INEQUALITY ... 26
1.19 THE CAUCHY-SCHWARZ INEQUALITY ... 27
1.20 PLUS AND MINUS INFINITY AND THE EXTENDED REAL NUMBER SYSTEM R* ... 28
1.21 COMPLEX NUMBERS ... 29
1.24 GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS ... 31
1.23 THE IMAGINARY UNIT ... 32
1.24 ABSOLUTE VALUE OF A COMPLEX NUMBER ... 32
1.25 IMPOSSIBILITY OF ORDERING THE COMPLEX NUMBERS ... 33
1.26 COMPLEX EXPONENTIALS ... 33
1.27 FURTHER PROPERTIES OF COMPLEX EXPONENTIALS ... 34
1.28 THE ARGUMENT OF A COMPLEX NUMBER ... 34
1.29 INTEGRAL POWERS AND ROOTS OF COMPLEX NUMBERS ... 35
1.30 COMPLEX LOGARITHMS ... 36
1.31 COMPLEX POWERS ... 37
1.32 COMPLEX SINES AND COSINES ... 38
1.33 INFINITY AND THE EXTENDED COMPLEX PLANE C* ... 38
EXERCISES ... 39
CHAPTER 2 SOME BASIC NOTIONS OF SET THEORY ... 46
2.1 INTRODUCTION ... 46
2.2 NOTATIONS ... 46
2.3 ORDERED PAIRS ... 47
2.4 CARTESIAN PRODUCT OF TWO SETS ... 47
2.5 RELATIONS AND FUNCTIONS ... 48
2.6 FURTHER TERMINOLOGY CONCERNING FUNCTIONS ... 49
2.7 ONE-TO-ONE FUNCTIONS AND INVERSES ... 50
2.8 COMPOSITE FUNCTIONS ... 51
2.9 SEQUENCES ... 51
2.10 SIMILAR (EQUINUMEROUS) SETS ... 52
2.11 FINITE AND INFINITE SETS ... 52
2.12 COUNTABLE AND UNCOUNTABLE SETS ... 53
2.13 UNCOUNTABILITY OF THE REAL NUMBER SYSTEM ... 53
2.14 SET ALGEBRA ... 54
2.15 COUNTABLE COLLECTIONS OF COUNTABLE SETS ... 56
EXERCISES ... 57
CHAPTER 3 ELEMENTS OF POINT SET TOPOLOGY ... 61
3.1 INTRODUCTION ... 61
3.2 EUCLIDEAN SPACE R^n ... 61
3.3 OPEN BALLS AND OPEN SETS IN R^n ... 63
3.4 THE STRUCTURE OF OPEN SETS IN R1 ... 64
3.5 CLOSED SETS ... 66
3.6 ADHERENT POINTS. ACCUMULATION POINTS ... 66
3.7 CLOSED SETS AND ADHERENT POINTS ... 67
3.8 THE BOLZANO-WEIERSTRASS THEOREM ... 68
3.9 THE CANTOR INTERSECTION THEOREM ... 70
3.10 THE LINDELOF COVERING THEOREM ... 70
3.11 THE HEENE-BOREL COVERING THEOREM ... 72
3.12 COMPACTNESS IN R^n ... 73
3.13 METRIC SPACES ... 74
3.14 POINT SET TOPOLOGY IN METRIC SPACES ... 75
3.15 COMPACT SUBSETS OF A METRIC SPACE ... 77
3.16 BOUNDARY OF A SET ... 78
EXERCISES ... 79
CHAPTER 4 LIMITS AND CONTINUITY ... 84
4.1 INTRODUCTION ... 84
4.2 CONVERGENT SEQUENCES IN A METRIC SPACE ... 84
4.3 CAUCHY SEQUENCES ... 86
4.4 COMPLETE METRIC SPACES ... 88
4.4 LIMIT OF A FUNCTION ... 88
4.6 LIMITS OF COMPLEX-VALUED FUNCTIONS ... 90
4.7 LIMITS OF VECTOR-VALUED FUNCTIONS ... 91
4.8 CONTINUOUS FUNCTIONS ... 92
4.9 CONTINUITY OF COMPOSITE FUNCTIONS ... 93
4.10 CONTINUOUS COMPLEX-VALUED AND VECTOR-VALUED FUNCTIONS ... 94
4.11 EXAMPLES OF CONTINUOUS FUNCTIONS ... 94
4.12 CONTINUITY AND INVERSE IMAGES OF OPEN OR CLOSED SETS ... 95
4.13 FUNCTIONS CONTINUOUS ON COMPACT SETS ... 96
4.14 TOPOLOGICAL MAPPINGS (HOMEOMORPHISMS) ... 98
4.15 BOLZANO'S THEOREM ... 98
4.16 CONNECTEDNESS ... 100
4.17 COMPONENTS OF A METRIC SPACE ... 101
4.18 ARCWISE CONNECTEDNESS ... 102
4.19 UNIFORM CONTINUITY ... 104
4.20 UNIFORM CONTINUITY AND COMPACT SETS ... 105
4.21 FIXED-POINT THEOREM FOR CONTRACTIONS ... 106
4.22 DISCONTINUITIES OF REAL-VALUED FUNCTIONS ... 106
4.23 MONOTONIC FUNCTIONS ... 108
EXERCISES ... 110
CHAPTER 5 DERIVATIVIS ... 118
5.1 INTRODUCTION ... 118
5.2 DEFINITION OF DERIVATIVE ... 118
5.3 DERIVATIVES AND CONTINUITY ... 119
5.4 ALGEBRA OF DERIVATIVES ... 120
5.6 ONE-SIDED DERIVATIVES AND INFINITE DERIVATIVES ... 121
5.7 FUNCTIONS WITH NONZERO DERIVATIVE ... 122
5.8 ZERO DERIVATIVES AND LOCAL EXTREMA ... 123
5.9 ROLLE'S THEOREM ... 124
5.10 THE MEAN-VALUE THEOREM FOR DERIVATIVES ... 124
5.11 INTERMEDIATE-VALUE THEOREM FOR DERIVATIVES ... 125
5.12 TAYLOR'S FORMULA WITH REMAINDER ... 127
5.13 DERIVATIVES OF VECTOR-VALUED FUNCTIONS ... 128
5.14 PARTIAL DERIVATIVES ... 129
5.15 DIFFERENTIATION OF FUNCTIONS OF A COMPLEX VARIABLE ... 130
5.16 THE CAUCHY-RIEMANN EQUATIONS ... 132
EXERCISES ... 135
CHAPTER 6 FUNCTIONS OF BOUNDED VARIATION AND RECTIFIABLE CURVES ... 141
6.1 INTRODUCTION ... 141
6.2 PROPERTIES^ MONOTONIC FUNCTIONS ... 141
6.3 FUNCTIONS OF BOUNDED VARIATION ... 142
6.4 TOTAL VARIATION ... 143
6.5 ADDITIVE PROPERTY OF TOTAL VARIATION ... 144
6.6 TOTAL VARIATION ON [a, x] AS A FUNCTION OF x ... 145
6.7 FUNCTIONS OF BOUNDED VARIATION EXPRESSED AS THE DIFFERENCE OF INCREASING FUNCTIONS ... 146
6.8 CONTINUOUS FUNCTIONS OF BOUNDED VARIATION ... 146
6.9 CURVES AND PATHS ... 147
6.10 RECTIFIABLE PATHS AND ARC LENGTH ... 148
6.11 ADDITIVE AND CONTINUITY PROPERTIES OF ARC LENGTH ... 149
6.12 EQUIVALENCE OF PATHS. CHANGE OF PARAMETER ... 150
EXERCISES ... 151
CHAPTER 7 THE RIEMANN-STIELTJES INTEGRAL ... 154
7.1 INTRODUCTION ... 154
7.2 NOTATION ... 155
7.4 LINEAR PROPERTIES ... 156
7.5 INTEGRATION BY PARTS ... 158
7.7 REDUCTION TO A RIEMANN INTEGRAL ... 159
7.8 STEP FUNCTIONS AS INTEGRATORS ... 161
7.9 REDUCTION OF A RIEMANN-STIELTJES INTEGRAL TO A FINITE SUM ... 162
7.10 EULER'S SUMMATION FORMULA ... 163
7.11 MONOTONICALLY INCREASING INTEGRATORS. UPPER AND LOWER INTEGRALS ... 164
7.12 ADDITIVE AND LINEARITY PROPERTIES OF UPPER AND LOWER INTEGRALS ... 167
7.13 RIEMANN'S CONDITION ... 167
7.14 COMPARISON THEOREMS ... 169
7.15 INTEGRATORS OF BOUNDED VARIATION ... 170
7.16 SUFFICIENT CONDITIONS FOR EXISTENCE OF RIEMANN-STTELTJES INTEGRALS ... 173
7.17 NECESSARY CONDITIONS FOR EXISTENCE OF RIEMANN-STIELTJES INTEGRALS ... 174
7.18 MEAN-VALUE THEOREMS FOR RIEMANN-STIELTJES INTEGRALS ... 174
7.19 THE INTEGRAL AS A FUNCTION OF THE INTERVAL ... 175
7.20 SECOND FUNDAMENTAL THEOREM OF INTEGRAL CALCULUS ... 176
7.22 SECOND MEAN-VALUE THEOREM FOR RIEMANN INTEGRALS ... 179
7.23 RIEMANN-STIELTJES INTEGRALS DEPENDING ON A PARAMETER ... 180
7.24 DIFFERENTIATION UNDER THE INTEGRAL SIGN ... 181
7.26 LEBESGUE'S CRITERION FOR EXISTENCE OF RIEMANN INTEGRALS ... 183
7.27 COMPLEX-VALUED RDEMANN-STIELTJES INTEGRALS ... 187
EXERCISES ... 188
CHAPTER 8 INFINITE SERIES AND INFINITE PRODUCTS ... 197
8.1 INTRODUCTION ... 197
8.2 CONVERGENT AND DIVERGENT SEQUENCES OF COMPLEX NUMBERS ... 197
8.3 LIMIT SUPERIOR AND LIMIT INFERIOR OF A REAL-VALUED SEQUENCE ... 198
8.4 MONOTONIC SEQUENCES OF REAL NUMBERS ... 199
8.5 INFINITE SERIES ... 199
8.6 INSERTING AND REMOVING PARENTHESES ... 201
8.7 ALTERNATING SERIES ... 202
8.8 ABSOLUTE AND CONDITIONAL CONVERGENCE ... 203
8.9 REAL AND IMAGINARY PARTS OF A COMPLEX SERIES ... 203
8.10 TESTS FOR CONVERGENCE OF SERIES WITH POSITIVE TERMS ... 204
8.11 THE GEOMETRIC SERIES ... 204
8.12 THE INTEGRAL TEST ... 205
8.13 THE BIG OH AND LITTLE OH NOTATION ... 206
8.14 THE RATIO TEST AND ROOT TEST ... 207
8.15 DIRICHLET'S TEST AND ABEL'S TEST. ... 207
8.16 PARTIAL SUMS OF THE GEOMETRIC SERIES "£zn ON THE UNIT CIRCLE |*| = 1 ... 209
8.17 REARRANGEMENTS OF SERIES ... 210
8.18 MEMANN'S THEOREM ON CONDITIONALLY CONVERGENT SERIES ... 211
8.19 SUBSERIES ... 211
8.20 DOUBLE SEQUENCES ... 213
8.21 DOUBLE SERIES ... 214
8.22 REARRANGEMENT THEOREM FOR DOUBLE SERIES ... 215
8.23 A SUFFICIENT CONDITION FOR EQUALITY OF ITERATED SERIES ... 216
8.24 MULTIPLICATION OF SERIES ... 217
8.25 CESARO SUMMABILITY ... 219
8.26 INFINITE PRODUCTS ... 220
8.27 EULER'S PRODUCT FOR THE RIEMANN ZETA FUNCTION ... 223
EXERCISES ... 224
CHAPTER 9 SEQUENCES OF FUNCTIONS ... 232
9.1 POINTWISE CONVERGENCE OF SEQUENCES OF FUNCTIONS ... 232
9.2 EXAMPLES OF SEQUENCES OF REAL-VALUED FUNCTIONS ... 233
9.3 DEFINITION OF UNIFORM CONVERGENCE ... 234
9.4 UNIFORM CONVERGENCE AND CONTINUITY ... 235
9.5 THE CAUCHY CONDITION FOR UNIFORM CONVERGENCE ... 236
9.6 UNIFORM CONVERGENCE OF INFINITE SERIES OF FUNCTIONS ... 237
9.7 A SPACE-FILLING CURVE ... 238
9.8 UNIFORM CONVERGENCE AND RIEMANN-STIELTJES INTEGRATION ... 239
9.9 NONUNIFORMLY CONVERGENT SEQUENCES THAT CAN BE INTEGRATED TERM BY TERM ... 240
9.10 UNIFORM CONVERGENCE AND DIFFERENTIATION ... 242
9.11 SUFFICIENT CONDITIONS FOR UNIFORM CONVERGENCE OF A SERIES ... 244
9.12 UNIFORM CONVERGENCE AND DOUBLE SEQUENCES ... 245
9.13 MEAN CONVERGENCE ... 246
9.14 POWER SERIES ... 248
9.15 MULTIPLICATION OF POWER SERIES ... 251
9.16 THE SUBSTITUTION THEOREM ... 252
9.17 RECIPROCAL OF A POWER SERIES ... 253
9.18 REAL POWER SERIES ... 254
9.19 THE TAYLOR'S SERIES GENERATED BY A FUNCTION ... 255
9.20 BERNSTEIN'S THEOREM ... 256
9.21 THE BINOMIAL SERIES ... 258
9.22 ABEL'S LIMIT THEOREM ... 258
9.23 TAUBER'S THEOREM ... 260
EXERCISES ... 261
CHAPTER 10 THE LEBESGUE INTEGRAL ... 266
10.1 INTRODUCTION ... 266
10.2 THE INTEGRAL OF A STEP FUNCTION ... 267
10.3 MONOTONIC SEQUENCES OF STEP FUNCTIONS ... 268
10.4 UPPER FUNCTIONS AND THEIR INTEGRALS ... 270
10.5 RIEMANN-INTEGRABLE FUNCTIONS AS EXAMPLES OF UPPER FUNCTIONS ... 273
10.6 THE CLASS OF LEBESGUE-INTEGRABLE FUNCTIONS ON A GENERAL INTERVAL ... 274
10.7 BASIC PROPERTIES OF THE LEBESGUE INTEGRAL ... 275
10.8 LEBESGUE INTEGRATION AND SETS OF MEASURE ZERO ... 278
10.9 THE LEVI MONOTONE CONVERGENCE THEOREMS ... 279
10.10 THE LEBESGUE DOMINATED CONVERGENCE THEOREM ... 284
10.11 APPLICATIONS OF LEBESGUE'S DOMINATED CONVERGENCE THEOREM ... 286
10.12 LEBESGUE INTEGRALS ON UNBOUNDED INTERVALS AS LIMITS OF INTEGRALS ON BOUNDED INTERVALS ... 288
10.13 IMPROPER RIEMANN INTEGRALS ... 290
10.14 MEASURABLE FUNCTIONS ... 293
10.15 CONTINUITY OF FUNCTIONS DEFINED BY LEBESGUE INTEGRALS ... 295
10.16 DIFFERENTIATION UNDER THE INTEGRAL SIGN ... 297
10.17 INTERCHANGING THE ORDER OF INTEGRATION ... 301
10.18 MEASURABLE SETS ON THE REAL LINE ... 303
10.19 THE LEBESGUE INTEGRAL OVER ARBITRARY SUBSETS OF R ... 305
10.20 LEBESGUE INTEGRALS OF COMPLEX-VALUED FUNCTIONS ... 306
10.21 INNER PRODUCTS AND NORMS ... 307
10.22 THE SET L^2(I) OF SQUARE-INTEGRABLE FUNCTIONS ... 308
10.23 THE SET L^2(I) AS A SEMIMETRIC SPACE ... 309
10.25 THE RIESZ-FISCHER THEOREM ... 311
EXERCISES ... 312
CHAPTER 11 FOURIER SERIES AND FOURIER INTEGRALS ... 320
11.1 INTRODUCTION ... 320
11.2 ORTHOGONAL SYSTEMS OF FUNCTIONS ... 320
11.3 THE THEOREM ON BEST APPROXIMATION ... 321
11.4 THE FOURIER SERIES OF A FUNCTION RELATIVE TO AN ORTHONORMAL SYSTEM ... 323
11.5 PROPERTIES OF THE FOURIER COEFFICIENTS ... 323
11.6 THE RIESZ-FISCHER THEOREM ... 326
11.7 THE CONVERGENCE AND REPRESENTATION PROBLEMS FOR TRIGONOMETRIC SERIES ... 326
11.8 THE RIEMANN-LEBESGUE LEMMA ... 327
11.9 THE DIRICHLET INTEGRALS ... 328
11.10 AN INTEGRAL REPRESENTATION FOR THE PARTIAL SUMS OF A FOURIER SERIES ... 331
11.11 REEMANN'S LOCALIZATION THEOREM ... 332
11.12 SUFFICIENT CONDITIONS FOR CONVERGENCE OF A FOURIER SERIES AT A PARTICULAR POINT ... 333
11.13 CESARO SUMMABILITY OF FOURIER SERIES ... 333
11.14 CONSEQUENCES OF FEJ&TS THEOREM ... 335
11.15 THE WEIERSTRASS APPROXIMATION THEOREM ... 336
11.16 OTHER FORMS OF FOURIER SERIES ... 336
11.17 THE FOURIER INTEGRAL THEOREM ... 337
11.18 THE EXPONENTIAL FORM OF THE FOURIER INTEGRAL THEOREM ... 339
11.19 INTEGRAL TRANSFORMS ... 340
11.20 CONVOLUTIONS ... 341
11.21 THE CONVOLUTION THEOREM FOR FOURIER TRANSFORMS ... 343
11.22 THE POISSON SUMMATION FORMULA ... 346
EXERCISES ... 349
CHAPTER 12 MULTIVARIABLE DIFFERENTIAL CALCULUS ... 358
12.1 INTRODUCTION ... 358
12.2 THE DIRECTIONAL DERIVATIVE ... 358
12.3 DIRECTIONAL DERIVATIVES AND CONTINUITY ... 359
12.4 THE TOTAL DERIVATIVE ... 360
12.5 THE TOTAL DERIVATIVE EXPRESSED IN TERMS OF PARTIAL DERIVATIVES ... 361
12.6 AN APPLICATION TO COMPLEX-VALUED FUNCTIONS ... 362
12.7 THE MATRIX OF A LINEAR FUNCTION ... 363
12.8 THE JACOBIAN MATRIX ... 365
12.9 THE CHAIN RULE ... 366
12.10 MATRIX FORM OF THE CHAIN RULE ... 367
12.11 THE MEAN-VALUE THEOREM FOR DIFFERENTIABLE FUNCTIONS ... 369
12.12 A SUFFICIENT CONDITION FOR DIFFERENTIABILITY ... 371
12.13 A SUFFICIENT CONDITION FOR EQUALITY OF MIXED PARTIAL DERIVATIVES ... 372
12.14 TAYLOR'S FORMULA FOR FUNCTIONS FROM R" TO R1 ... 375
EXERCISES ... 376
CHAPTER 13 IMPLICIT FUNCTIONS AND EXTREMUM PROBLEMS ... 381
13.1 INTRODUCTION ... 381
13.2 FUNCTIONS WITH NONZERO JACOBIAN DETERMINANT ... 381
13.3 THE INVERSE FUNCTION THEOREM ... 386
13.4 THE IMPLICIT FUNCTION THEOREM ... 387
13.5 EXTREMA OF REAL-VALUED FUNCTIONS OF ONE VARIABLE ... 389
13.6 EXTREMA OF REAL-VALUED FUNCTIONS OF SEVERAL VARIABLES ... 390
13.7 EXTREMUM PROBLEMS WITH SIDE CONDITIONS ... 394
EXERCISES ... 398
CHAPTER 14 MULTIPLE RIEMANN INTEGRALS ... 402
14.1 INTRODUCTION ... 402
14.2 THE MEASURE OF A BOUNDED INTERVAL IN R" ... 402
14.3 THE RIEMANN INTEGRAL OF A BOUNDED FUNCTION DEFINED ON A COMPACT INTERVAL IN R" ... 403
14.4 SETS OF MEASURE ZERO AND LEBESGUE'S CRITERION FOR EXISTENCE OF A MULTIPLE RIEMANN INTEGRAL ... 405
14.5 EVALUATION OF A MULTIPLE INTEGRAL BY ITERATED INTEGRATION ... 405
14.6 JORDAN-MEASURABLE SETS IN Rn ... 410
14.7 MULTIPLE INTEGRATION OVER JORDAN-MEASURABLE SETS ... 411
14.8 JORDAN CONTENT EXPRESSED AS A RIEMANN INTEGRAL ... 412
14.9 ADDITIVE PROPERTY OF THE RIEMANN INTEGRAL ... 413
14.10 MEAN-VALUE THEOREM FOR MULTIPLE INTEGRALS ... 414
EXERCISES ... 416
CHAPTER 15 MULTIPLE LEBESGUE INTEGRALS ... 419
15.1 INTRODUCTION ... 419
15.2 STEP FUNCTIONS AND THEIR INTEGRALS ... 420
15.3 UPPER FUNCTIONS AND LEBESGUE-INTEGRABLE FUNCTIONS ... 420
15.4 MEASURABLE FUNCTIONS AND MEASURABLE SETS IN R^n ... 421
15.5 FUBINPS REDUCTION THEOREM FOR THE DOUBLE INTEGRAL OF A STEP FUNCTION ... 423
15.6 SOME PROPERTIES OF SETS OF MEASURE ZERO ... 425
15.7 FUBINI'S REDUCTION THEOREM FOR DOUBLE INTEGRALS ... 427
15.8 THE TONELU-HOBSON TEST FOR INTEGRABILITY ... 429
15.9 COORDINATE TRANSFORMATIONS ... 430
15.10 THE TRANSFORMATION FORMULA FOR MULTIPLE INTEGRALS ... 435
15.11 PROOF OF THE TRANSFORMATION FORMULA FOR LINEAR COORDINATE TRANSFORMATIONS ... 435
15.12 PROOF OF THE TRANSFORMATION FORMULA FOR THE CHARACTERISTIC FUNCTION OF A COMPACT CUBE ... 437
15.13 COMPLETION OF THE PROOF OF THE TRANSFORMATION FORMULA ... 443
EXERCISES ... 444
CHAPTER 16 CAUCHY'S THEOREM AND THE RESIDUE CALCULUS ... 448
16.1 ANALYTIC FUNCTIONS ... 448
16.2 PATHS AND CURVES IN THE COMPLEX PLANE ... 449
16.3 CONTOUR INTEGRALS ... 450
16.4 THE INTEGRAL ALONG A CIRCULAR PATH AS A FUNCTION OF THE RADIUS ... 452
16.5 CAUCHY'S INTEGRAL THEOREM FOR A CIRCLE ... 453
16.6 HOMOTOPIC CURVES ... 453
16.7 INVARIANCE OF CONTOUR INTEGRALS UNDER HOMOTOPY ... 456
16.8 GENERAL FORM OF CAUCHY'S INTEGRAL THEOREM ... 457
16.9 CAUCHY'S INTEGRAL FORMULA ... 457
16.10 THE WINDING NUMBER OF A CIRCUIT WITH RESPECT TO A POINT ... 458
16.11 THE UNBOUNDEDNESS OF THE SET OF POINTS WITH WINDING NUMBER ZERO ... 460
16.12 ANALYTIC FUNCTIONS DEFINED BY CONTOUR INTEGRALS ... 461
16.13 POWER-SERIES EXPANSIONS FOR ANALYTIC FUNCTIONS ... 463
16.15 ISOLATION OF THE ZEROS OF AN ANALYTIC FUNCTION ... 465
16.16 THE IDENTITY THEOREM FOR ANALYTIC FUNCTIONS ... 466
16.17 THE MAXIMUM AND MINIMUM MODULUS OF AN ANALYTIC FUNCTION ... 467
16.18 THE OPEN MAPPING THEOREM ... 468
16.19 LAURENT EXPANSIONS FOR FUNCTIONS ANALYTIC IN AN ANNULUS ... 469
16.20 ISOLATED SINGULARITIES ... 471
16.21 THE RESIDUE OF A FUNCTION AT AN ISOLATED SINGULAR POINT ... 473
16.22 THE CAUCHY RESIDUE THEOREM ... 474
16.23 COUNTING ZEROS AND POLES IN A REGION ... 475
16.24 EVALUATION OF REAL-VALUED INTEGRALS BY MEANS OF RESIDUES ... 476
16.25 EVALUATION OF GAUSS'S SUM BY RESIDUE CALCULUS ... 478
16.26 APPLICATION OF THE RESIDUE THEOREM TO THE INVERSION FORMULA FOR LAPLACE TRANSFORMS ... 482
16.27 CONFORMAL MAPPINGS ... 484
EXERCISES ... 486
INDEX OF SPECIAL SYMBOLS ... 498
INDEX ... 499