Handbook of complex variables
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| Sprache: | English |
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Boston [u.a.]
Birkhäuser
1999
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| 100 | 1 | |a Krantz, Steven G. |d 1951- |e Verfasser |0 (DE-588)130535907 |4 aut | |
| 245 | 1 | 0 | |a Handbook of complex variables |c Steven G. Krantz |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 1999 | |
| 300 | |a XXIV, 290 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Funktionentheorie - Anwendung | |
| 650 | 4 | |a Komplexe Funktion | |
| 650 | 0 | 7 | |a Funktionentheorie |0 (DE-588)4018935-1 |2 gnd |9 rswk-swf |
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| adam_text | IMAGE 1
STEVEN G. KRANTZ
HANDBOOK OF COMPLEX VARIABLES
WITH 102 FIGURES
BIRKHAEUSER BOSTON * BASEL * BERLIN
IMAGE 2
CONTENTS
PREFACE XIX
LIST OF FIGURES XXI
1 THE COMPLEX PLANE 1
1.1 COMPLEX ARITHMETIC 1
1.1.1 THE REAL NUMBERS 1
1.1.2 THE COMPLEX NUMBERS . 1
1.1.3 COMPLEX CONJUGATE 2
1.1.4 MODULUS OF A COMPLEX NUMBER 2
1.1.5 THE TOPOLOGY OF THE COMPLEX PLANE 3
1.1.6 THE COMPLEX NUMBERS AS A FIELD 6
1.1.7 THE FUNDAMENTAL THEOREM OF ALGEBRA 7
1.2 THE EXPONENTIAL AND APPLICATIONS 7
1.2.1 THE EXPONENTIAL FUNCTION 7
1.2.2 THE EXPONENTIAL USING POWER SERIES 8
1.2.3 LAWS OF EXPONENTIATION 8
1.2.4 POLAR FORM OF A COMPLEX NUMBER 8
1.2.5 ROOTS OF COMPLEX NUMBERS 10
1.2.6 THE ARGUMENT OF A COMPLEX NUMBER 11
1.2.7 FUNDAMENTAL INEQUALITIES 12
1.3 HOLOMORPHIC FUNCTIONS 12
1.3.1 CONTINUOUSLY DIFFERENTIABLE AND C K FUNCTIONS . . 12
1.3.2 THE CAUCHY-RIEMANN EQUATIONS 13
1.3.3 DERIVATIVES 13
1.3.4 DEFINITION OF HOLOMORPHIC FUNCTION 14
1.3.5 THE COMPLEX DERIVATIVE 15
VII
IMAGE 3
VLLL
1.3.6 ALTERNATIVE TERMINOLOGY FOR
HOLOMORPHIC FUNCTIONS 16
1.4 THE RELATIONSHIP OF HOLOMORPHIC AND HARMONIE FUNCTIONS . 16
1.4.1 HARMONIE FUNCTIONS 16
1.4.2 HOLOMORPHIC AND HARMONIE FUNCTIONS 17
2 COMPLEX LINE INTEGRALS 19
2.1 REAL AND COMPLEX LINE INTEGRALS 19
2.1.1 CURVES 19
2.1.2 CLOSED CURVES 19
2.1.3 DIFFERENTIABLE AND C K CURVES 21
2.1.4 INTEGRALS ON CURVES 21
2.1.5 THE FUNDAMENTAL THEOREM OF CALCULUS ALONG CURVES 22
2.1.6 THE COMPLEX LINE INTEGRAL 22
2.1.7 PROPERTIES OF INTEGRALS 22
2.2 COMPLEX DIFFERENTIABILITY AND CONFORMALITY 23
2.2.1 LIMITS 23
2.2.2 CONTINUITY 24
2.2.3 THE COMPLEX DERIVATIVE 24
2.2.4 HOLOMORPHICITY AND THE COMPLEX DERIVATIVE . .. 24
2.2.5 CONFORMALITY 25
2.3 THE CAUCHY INTEGRAL THEOREM AND FORMULA 26
2.3.1 THE CAUCHY INTEGRAL FORMULA 26
2.3.2 THE CAUCHY INTEGRAL THEOREM, BASIC FORM . . .. 26
2.3.3 MORE GENERAL FORMS OF THE CAUCHY THEOREMS . . 26
2.3.4 DEFORMABILITY OF CURVES 28
2.4 A CODA ON THE LIMITATIONS OF THE CAUCHY INTEGRAL FORMULA 28
3 APPLICATIONS OF THE CAUCHY THEORY 31
3.1 THE DERIVATIVES OF A HOLOMORPHIC FUNCTION 31
3.1.1 A FORMULA FOR THE DERIVATIVE 31
3.1.2 THE CAUCHY ESTIMATES 31
3.1.3 ENTIRE FUNCTIONS AND LIOUVILLE S THEOREM 31
3.1.4 THE FUNDAMENTAL THEOREM OF ALGEBRA 32
3.1.5 SEQUENCES OF HOLOMORPHIC FUNCTIONS AND THEIR DERIVATIVES 33
3.1.6 THE POWER SERIES REPRESENTATION OF A HOLOMORPHIC FUNCTION 34
3.1.7 TABLE OF ELEMENTARY POWER SERIES 35
3.2 THE ZEROS OF A HOLOMORPHIC FUNCTION 36
3.2.1 THE ZERO SET OF A HOLOMORPHIC FUNCTION 36
IMAGE 4
CONTENTS IX
3.2.2 DISCRETE SETS AND ZERO SETS 37
3.2.3 UNIQUENESS OF ANALYTIC CONTINUATION 38
4 ISOLATED SINGULARITIES AND L A U R E NT SERIES 41
4.1 THE BEHAVIOR OF A HOLOMORPHIC FUNCTION NEAR AN ISOLATED SINGULARITY
41
4.1.1 ISOLATED SINGULARITIES 41
4.1.2 A HOLOMORPHIC FUNCTION ON A PUNCTURED DOMAIN 41
4.1.3 CLASSIFICATION OF SINGULARITIES 41
4.1.4 REMOVABLE SINGULARITIES, POLES, AND ESSENTIAL SINGULARITIES 42
4.1.5 THE RIEMANN REMOVABLE SINGULARITIES THEOREM 42
4.1.6 THE CASORATI-WEIERSTRASS THEOREM 43
4.2 EXPANSION AROUND SINGULAR POINTS 43
4.2.1 LAURENT SERIES 43
4.2.2 CONVERGENCE OF A DOUBLY INFINITE SERIES 43
4.2.3 ANNULUS OF CONVERGENCE 44
4.2.4 UNIQUENESS OF THE LAURENT EXPANSION 44
4.2.5 THE CAUCHY INTEGRAL FORMULA FOR AN ANNULUS . .. 45
4.2.6 EXISTENCE OF LAURENT EXPANSIONS 45
4.2.7 HOLOMORPHIC FUNCTIONS WITH ISOLATED SINGULARITIES 45
4.2.8 CLASSIFICATION OF SINGULARITIES IN TERMS OF LAURENT SERIES . 46
4.3 EXAMPLES OF LAURENT EXPANSIONS 46
4.3.1 PRINCIPAL PART OF A FUNCTION 46
4.3.2 ALGORITHM FOR CALCULATING THE COEFFICIENTS OF THE LAURENT
EXPANSION 48
4.4 THE CALCULUS OF RESIDUES 48
4.4.1 FUNCTIONS WITH MULTIPLE SINGULARITIES 48
4.4.2 THE RESIDUE THEOREM 48
4.4.3 RESIDUES 49
4.4.4 THE INDEX OR WINDING NUMBER OF A CURVE ABOUT A POINT 49
4.4.5 RESTATEMENT OF THE RESIDUE THEOREM 50
4.4.6 METHOD FOR CALCULATING RESIDUES 50
4.4.7 SUMMARY CHARTS OF LAURENT SERIES AND RESIDUES . 51 4.5
APPLICATIONS TO THE CALCULATION OF DEFINITE INTEGRALS AND SUMS 51
4.5.1 THE EVALUATION OF DEFINITE INTEGRALS 51
4.5.2 A BASIC EXAMPLE 52
4.5.3 COMPLEXIFICATION OF THE INTEGRAND 54
IMAGE 5
X
CONTENTS
4.5.4 AN EXAMPLE WITH A MORE SUBTLE CHOICE OFCONTOUR 56
4.5.5 MAKING THE SPURIOUS PART OF THE INTEGRAL DISAPPEAR 58
4.5.6 THE USE OF THE LOGARITHM 60
4.5.7 SUMMING A SERIES USING RESIDUES 62
4.5.8 SUMMARY CHART OF SOME INTEGRATION TECHNIQUES 63
4.6 MEROMORPHIC FUNCTIONS AND SINGULARITIES AT INFINITY 63
4.6.1 MEROMORPHIC FUNCTIONS 63
4.6.2 DISCRETE SETS AND ISOLATED POINTS 63
4.6.3 DEFINITION OF MEROMORPHIC FUNCTION 64
4.6.4 EXAMPLES OF MEROMORPHIC FUNCTIONS 64
4.6.5 MEROMORPHIC FUNCTIONS WITH INFINITELY MANY POLES 66
4.6.6 SINGULARITIES AT INFINITY 66
4.6.7 THE LAURENT EXPANSION AT INFINITY 67
4.6.8 MEROMORPHIC AT INFINITY 67
4.6.9 MEROMORPHIC FUNCTIONS IN THE EXTENDED PLANE 67
THE ARGUMENT PRINCIPLE 69
5.1 COUNTING ZEROS AND POLES 69
5.1.1 LOCAL GEOMETRIE BEHAVIOR OF A HOLOMORPHIC FUNCTION 69
5.1.2 LOCATING THE ZEROS OF A HOLOMORPHIC FUNCTION . . 69 5.1.3 ZEROOF
ORDERN 70
5.1.4 COUNTING THE ZEROS OF A HOLOMORPHIC FUNCTION . . 70 5.1.5 THE
ARGUMENT PRINCIPLE 71
5.1.6 LOCATION OF POLES 72
5.1.7 THE ARGUMENT PRINCIPLE FOR MEROMORPHIC FUNCTIONS 72
5.2 THE LOCAL GEOMETRY OF HOLOMORPHIC FUNCTIONS 73
5.2.1 THE OPEN MAPPING THEOREM 73
5.3 FURTHER RESULTS ON THE ZEROS OF HOLOMORPHIC FUNCTIONS . . 74 5.3.1
ROUCHE S THEOREM 74
5.3.2 TYPICAL APPLICATION OF ROUCHE S THEOREM . . .. 74
5.3.3 ROUCHE S THEOREM AND THE FUNDAMENTAL THEOREM OF ALGEBRA 75
5.3.4 HURWITZ S THEOREM 76
5.4 THE MAXIMUM PRINCIPLE 76
5.4.1 THE MAXIMUM MODULUS PRINCIPLE 76
5.4.2 BOUNDARY MAXIMUM MODULUS THEOREM . . . .. 76
IMAGE 6
CONTENTS
XI
5.4.3 THE MINIMUM PRINCIPLE 77
5.4.4 THE MAXIMUM PRINCIPLE ON AN UNBOUNDED DOMAIN 77
5.5 THE SCHWARZ LEMMA 77
5.5.1 SCHWARZ S LEMMA 78
5.5.2 THE SCHWARZ-PICK LEMMA 78
6 THE GEOMETRIE THEORY OF HOLOMORPHIC FUNCTIONS 79
6.1 THE IDEA OF A CONFORMAL MAPPING 79
6.1.1 CONFORMAL MAPPINGS 79
6.1.2 CONFORMAL SELF-MAPS OF THE PLANE 79
6.2 CONFORMAL MAPPINGS OF THE UNIT DISC 80
6.2.1 CONFORMAL SELF-MAPS OF THE DISC 80
6.2.2 MOEBIUS TRANSFORMATIONS 81
6.2.3 SELF-MAPS OF THE DISC 81
6.3 LINEAR FRACTIONAL TRANSFORMATIONS 81
6.3.1 LINEAR FRACTIONAL MAPPINGS 81
6.3.2 THE TOPOLOGY OF THE EXTENDED PLANE 83
6.3.3 THE RIEMANN SPHERE 83
6.3.4 CONFORMAL SELF-MAPS OF THE RIEMANN SPHERE . .. 84
6.3.5 THE CAYLEY TRANSFORM 85
6.3.6 GENERALIZED CIRCLES AND LINES 85
6.3.7 THE CAYLEY TRANSFORM REVISITED 85
6.3.8 SUMMARY CHART OF LINEAR FRACTIONAL TRANSFORMATIONS 85
6.4 THE RIEMANN MAPPING THEOREM 86
6.4.1 THE CONCEPT OF HOMEOMORPHISM 86
6.4.2 THE RIEMANN MAPPING THEOREM 86
6.4.3 THE RIEMANN MAPPING THEOREM: SECOND FORMULATION 87
6.5 CONFORMAL MAPPINGS OF ANNULI 87
6.5.1 A RIEMANN MAPPING THEOREM FOR ANNULI 87
6.5.2 CONFORMAL EQUIVALENCE OF ANNULI 87
6.5.3 CLASSIFICATION OF PLANAR DOMAINS 88
7 HARMONIE FUNCTIONS 89
7.1 BASIC PROPERTIES OF HARMONIE FUNCTIONS 89
7.1.1 THE LAPLACE EQUATION 89
7.1.2 DEFINITION OF HARMONIE FUNCTION 89
7.1.3 REAL- AND COMPLEX-VALUED HARMONIE FUNCTIONS 89
7.1.4 HARMONIE FUNCTIONS AS THE REAL PARTS OF HOLOMORPHIC FUNCTIONS 90
7.1.5 SMOOTHNESS OF HARMONIE FUNCTIONS 90
IMAGE 7
XII
CONTENTS
7.2 THE MAXIMUM PRINCIPLE AND THE MEAN VALUE PROPERTY 91
7.2.1 THE MAXIMUM PRINCIPLE FOR HARMONIE FUNCTIONS 91
7.2.2 THE MINIMUM PRINCIPLE FOR HARMONIE FUNCTIONS 91
7.2.3 THE BOUNDARY MAXIMUM AND MINIMUM PRINCIPLES 91
7.2.4 THE MEAN VALUE PROPERTY 92
7.2.5 BOUNDARY UNIQUENESS FOR HARMONIE FUNCTIONS . . 92 7.3 THE POISSON
INTEGRAL FORMULA 92
7.3.1 THE POISSON INTEGRAL 92
7.3.2 THE POISSON KERNEL 93
7.3.3 THE DIRICHLET PROBLEM 93
7.3.4 THE SOLUTION OF THE DIRICHLET PROBLEM ON THE DISC 93
7.3.5 THE DIRICHLET PROBLEM ON A GENERAL DISC 94
7.4 REGULARITY OF HARMONIE FUNCTIONS 94
7.4.1 THE MEAN VALUE PROPERTY ON CIRCLES 94
7.4.2 THE LIMIT OF A SEQUENCE OF HARMONIE FUNCTIONS . 95
7.5 THE SCHWARZ REFLECTION PRINCIPLE 95
7.5.1 REFLECTION OF HARMONIE FUNCTIONS 95
7.5.2 SCHWARZ REFLECTION PRINCIPLE FOR HARMONIE FUNCTIONS 95
7.5.3 THE SCHWARZ REFLECTION PRINCIPLE FOR HOLOMORPHIC FUNCTIONS 96
7.5.4 MORE GENERAL VERSIONS OF THE SCHWARZ REFLECTION PRINCIPLE 96
7.6 HARNACK S PRINCIPLE 97
7.6.1 THE HARNACK INEQUALITY 97
7.6.2 HARNACK S PRINCIPLE 97
7.7 THE DIRICHLET PROBLEM AND SUBHARMONIC FUNCTIONS 97
7.7.1 THE DIRICHLET PROBLEM 97
7.7.2 CONDITIONS FOR SOLVING THE DIRICHLET PROBLEM . .. 98
7.7.3 MOTIVATION FOR SUBHARMONIC FUNCTIONS 98
7.7.4 DEFINITION OF SUBHARMONIC FUNCTION 99
7.7.5 OTHER CHARACTERIZATIONS OF SUBHARMONIC FUNCTIONS 99
7.7.6 THE MAXIMUM PRINCIPLE 100
7.7.7 LACK OF A MINIMUM PRINCIPLE 100
7.7.8 BASIC PROPERTIES OF SUBHARMONIC FUNCTIONS . . .. 100
7.7.9 THE CONCEPT OF A BARRIER 100
7.8 THE GENERAL SOLUTION OF THE DIRICHLET PROBLEM 101
IMAGE 8
CONTENTS XIII
7.8.1 ENUNCIATION OF THE SOLUTION OF THE DIRICHLET PROBLEM 101
8 INFINITE SERIES AND PRODUCTS 103
8.1 BASIC CONCEPTS CONCERNING INFINITE SUMS AND PRODUCTS 103
8.1.1 UNIFORM CONVERGENCE OF A SEQUENCE 103
8.1.2 THE CAUCHY CONDITION FOR A SEQUENCE OF FUNCTIONS 103
8.1.3 NORMAL CONVERGENCE OF A SEQUENCE 103
8.1.4 NORMAL CONVERGENCE OF A SERIES 104
8.1.5 THE CAUCHY CONDITION FOR A SERIES 104
8.1.6 THE CONCEPT OF AN INFINITE PRODUCT 104
8.1.7 INFINITE PRODUCTS OF SEALARS 105
8.1.8 PARTIAL PRODUCTS 105
8.1.9 CONVERGENCE OF AN INFINITE PRODUCT 105
8.1.10 THE VALUE OF AN INFINITE PRODUCT 106
8.1.11 PRODUCTS THAT ARE DISALLOWED 106
8.1.12 CONDITION FOR CONVERGENCE OF AN INFINITE PRODUCT 106
8.1.13 INFINITE PRODUCTS OF HOLOMORPHIC FUNCTIONS . . .. 107 8.1.14
VANISHING OF AN INFINITE PRODUCT . 108
8.1.15 UNIFORM CONVERGENCE OF AN INFINITE PRODUCT OF FUNCTIONS 108
8.1.16 CONDITION FOR THE UNIFORM CONVERGENCE OF AN INFINITE PRODUCT OF
FUNCTIONS 108
8.2 THE WEIERSTRASS FACTORIZATION THEOREM 109
8.2.1 PROLOGUE 109
8.2.2 WEIERSTRASS FACTORS 109
8.2.3 CONVERGENCE OF THE WEIERSTRASS PRODUCT 110
8.2.4 EXISTENCE OF AN ENTIRE FUNCTION WITH PRESCRIBED ZEROS 110
8.2.5 THE WEIERSTRASS FACTORIZATION THEOREM 110
8.3 THE THEOREMS OF WEIERSTRASS AND MITTAG-LEFFLER 110
8.3.1 THE CONCEPT OF WEIERSTRASS S THEOREM 110
8.3.2 WEIERSTRASS S THEOREM 111
8.3.3 CONSTRUCTION OF A DISCRETE SET 111
8.3.4 DOMAINS OF EXISTENCE FOR HOLOMORPHIC FUNCTIONS 112
8.3.5 THE FIELD GENERATED BY THE RING OF HOLOMORPHIC FUNCTIONS 112
8.3.6 THE MITTAG-LEFFLER THEOREM 112
8.3.7 PRESCRIBING PRINCIPAL PARTS 113
IMAGE 9
XIV CONTENTS
8.4 NORMAL FAMILIES 113
8.4.1 NORMAL CONVERGENCE 113
8.4.2 NORMAL FAMILIES 114
8.4.3 MONTEL S THEOREM, FIRST VERSION 114
8.4.4 MONTEL S THEOREM, SECOND VERSION 114
8.4.5 EXAMPLES OF NORMAL FAMILIES 114
9 APPLICATIONS OF INFINITE SUMS AND PRODUCTS 117
9.1 JENSEN S FORMULA AND AN INTRODUCTION TO BLASCHKE PRODUCTS 117
9.1.1 BLASHKE FACTORS 117
9.1.2 JENSEN S FORMULA 117
9.1.3 JENSEN S INEQUALITY 118
9.1.4 ZEROS OF A BOUNDED HOLOMORPHIC FUNCTION . . .. 118
9.1.5 THE BLASCHKE CONDITION 118
9.1.6 BLASCHKE PRODUCTS 119
9.1.7 BLASCHKE FACTORIZATION 119
9.2 THE HADAMARD GAP THEOREM 119
9.2.1 THE TECHNIQUE OF OSTROWSKI 119
9.2.2 THE OSTROWSKI-HADAMARD GAP THEOREM 119
9.3 ENTIRE FUNCTIONS OF FINITE ORDER 120
9.3.1 RATE OF GROWTH AND ZERO SET 120
9.3.2 FINITE ORDER 121
9.3.3 FINITE ORDER AND THE EXPONENTIAL TERM OF WEIERSTRASS 121
9.3.4 WEIERSTRASS CANONICAL PRODUCTS 121
9.3.5 THE HADAMARD FACTORIZATION THEOREM 121
9.3.6 VALUE DISTRIBUTION THEORY . 122
10 ANALYTIC CONTINUATION 123
10.1 DEFINITION OF AN ANALYTIC FUNCTION ELEMENT 123
10.1.1 CONTINUATION OF HOLOMORPHIC FUNCTIONS 123
10.1.2 EXAMPLES OF ANALYTIC CONTINUATION 123
10.1.3 FUNCTION ELEMENTS 128
10.1.4 DIRECT ANALYTIC CONTINUATION 128
10.1.5 ANALYTIC CONTINUATION OF A FUNCTION 128
10.1.6 GLOBAL ANALYTIC FUNCTIONS 129
10.1.7 AN EXAMPLE OF ANALYTIC CONTINUATION 130
10.2 ANALYTIC CONTINUATION ALONG A CURVE 130
10.2.1 CONTINUATION ON A CURVE 130
10.2.2 UNIQUENESS OF CONTINUATION ALONG A CURVE . . .. 131
10.3 THE MONODROMY THEOREM 131
10.3.1 UNAMBIGUITY OF ANALYTIC CONTINUATION 132
10.3.2 THE CONCEPT OF HOMOTOPY 132
IMAGE 10
CONTENTS XV
10.3.3 FIXED ENDPOINT HOMOTOPY 133
10.3.4 UNRESTRICTED CONTINUATION 134
10.3.5 THE MONODROMY THEOREM 134
10.3.6 MONODROMY AND GLOBALLY DEFINED ANALYTIC FUNCTIONS 134
10.4 THE IDEA OF A RIEMANN SURFACE 135
10.4.1 WHAT IS A RIEMANN SURFACE? 135
10.4.2 EXAMPLES OF RIEMANN SURFACES 135
10.4.3 THE RIEMANN SURFACE FOR THE SQUARE ROOT FUNCTION 137
10.4.4 HOLOMORPHIC FUNCTIONS ON A RIEMANN SURFACE . . 137 10.4.5 THE
RIEMANN SURFACE FOR THE LOGARITHM 138
10.4.6 RIEMANN SURFACES IN GENERAL 139
10.5 PICARD S THEOREMS 140
10.5.1 VALUE DISTRIBUTION FOR ENTIRE FUNCTIONS 140
10.5.2 PICARD S LITTLE THEOREM 140
10.5.3 PICARD S GREAT THEOREM 140
10.5.4 THE LITTLE THEOREM, THE GREAT THEOREM, AND THE
CASORATI-WEIERSTRASS THEOREM 140
11 RATIONAL APPROXIMATION THEORY 143
11.1 RUNGE S THEOREM 143
11.1.1 APPROXIMATION BY RATIONAL FUNCTIONS 143
11.1.2 RUNGE S THEOREM 143
11.1.3 APPROXIMATION BY POLYNOMIALS 144
11.1.4 APPLICATIONS OF RUNGE S THEOREM 144
11.2 MERGELYAN S THEOREM 146
11.2.1 AN IMPROVEMENT OF RUNGE S THEOREM 146
11.2.2 A SPECIAL CASE OF MERGELYAN S THEOREM 146
11.2.3 GENERALIZED MERGELYAN THEOREM 147
12 SPECIAL CLASSES OF HOLOMORPHIC FUNCTIONS 149
12.1 SCHLICHT FUNCTIONS AND THE BIEBERBACH CONJECTURE 149
12.1.1 SCHLICHT FUNCTIONS 149
12.1.2 THE BIEBERBACH CONJECTURE 149
12.1.3 THE LUSIN AREA INTEGRAL 150
12.1.4 THE AREA PRINCIPLE 150
12.1.5 THE KOEBE 1/4 THEOREM 150
12.2 EXTENSION TO THE BOUNDARY OF CONFORMAL MAPPINGS 151
12.2.1 BOUNDARY CONTINUATION 151
12.2.2 SOME EXAMPLES CONCERNING BOUNDARY CONTINUATION 151
IMAGE 11
XVI CONTENTS
12.3 HARDY SPACES 152
12.3.1 THE DEFINITION OF HARDY SPACES 152
12.3.2 THE BLASCHKE FACTORIZATION FOR H 153
12.3.3 MONOTONICITY OF THE HARDY SPACE NORM 153
12.3.4 CONTAINMENT RELATIONS AMONG HARDY SPACES 153
12.3.5 THE ZEROS OF HARDY FUNCTIONS 153
12.3.6 THE BLASCHKE FACTORIZATION FOR H P FUNCTIONS . .. 154
13 SPECIAL FUNCTIONS 155
13.0 INTRODUCTION 155
13.1 THE GAMMA AND BETA FUNCTIONS 155
13.1.1 DEFINITION OF THE GAMMA FUNCTION 155
13.1.2 RECURSIVE IDENTITY FOR THE GAMMA FUNCTION . . .. 156
13.1.3 HOLOMORPHICITY OF THE GAMMA FUNCTION 156
13.1.4 ANALYTIC CONTINUATION OF THE GAMMA FUNCTION . . 156 13.1.5
PRODUCT FORMULA FOR THE GAMMA FUNCTION . . .. 156
13.1.6 NON-VANISHING OF THE GAMMA FUNCTION 156
13.1.7 THE EULER-MASCHERONI CONSTANT 156
13.1.8 FORMULA FOR THE RECIPROCAL OF THE GAMMA FUNCTION 157
13.1.9 CONVEXITY OF THE GAMMA FUNCTION 157
13.1.10 THE BOHR-MOLLERUP THEOREM 157
13.1.11 THE BETA FUNCTION 157
13.1.12 SYMMETRY OF THE BETA FUNCTION 158
13.1.13 RELATION OF THE BETA FUNCTION TO THE GAMMA FUNCTION 158
13.1.14 INTEGRAL REPRESENTATION OF THE BETA FUNCTION . . . 158 13.2
RIEMANN S ZETA FUNCTION 158
13.2.1 DEFINITION OF THE ZETA FUNCTION 158
13.2.2 THE EULER PRODUCT FORMULA 158
13.2.3 RELATION OF THE ZETA FUNCTION TO THE GAMMA FUNCTION 159
13.2.4 THE HANKEL CONTOUR AND HANKEL FUNCTIONS . . .. 159
13.2.5 EXPRESSION OF THE ZETA FUNCTION AS A HANKEL INTEGRAL 160
13.2.6 LOCATION OF THE POLE OF THE ZETA FUNCTION 160
13.2.7 THE FUNCTIONAL EQUATION 160
13.2.8 ZEROS OF THE ZETA FUNCTION 161
13.2.9 THE RIEMANN HYPOTHESIS 161
13.2.10 THE LAMBDA FUNCTION 161
13.2.11 RELATION OF THE ZETA FUNCTION TO THE LAMBDA FUNCTION 161
13.2.12 MORE ON THE ZEROS OF THE ZETA FUNCTION 162
IMAGE 12
CONTENTS
XVII
13.2.13 ZEROS OF THE ZETA FUNCTION AND THE BOUNDARY OF THE CRITICAL
STRIP 162
13.3 SOME COUNTING FUNCTIONS AND A FEW TECHNICAL LEMMAS . . 162 13.3.1
THE COUNTING FUNCTIONS OF CLASSICAL NUMBER THEORY 162
13.3.2 THE FUNCTION TT 162
13.3.3 THE PRIME NUMBER THEOREM 162
14 APPLICATIONS THAT DEPEND ON CONFORMAL MAPPING 163
14.1 CONFORMAL MAPPING 163
14.1.1 A LIST OF USEFUL CONFORMAL MAPPINGS 163
14.2 APPLICATION OF CONFORMAL MAPPING TO THE DIRICHLET PROBLEM 164
14.2.1 THE DIRICHLET PROBLEM 164
14.2.2 PHYSICAL MOTIVATION FOR THE DIRICHLET PROBLEM . . 164 14.3
PHYSICAL EXAMPLES SOLVED BY MEANS OF CONFORMAL MAPPING 168
14.3.1 STEADY STATE HEAT DISTRIBUTION ON A LENS-SHAPED REGION 169
14.3.2 ELECTROSTATICS ON A DISC 170
14.3.3 INCOMPRESSIBLE FLUID FLOW AROUND A POST 172
14.4 NUMERICAL TECHNIQUES OF CONFORMAL MAPPING 175
14.4.1 NUMERICAL APPROXIMATION OF THE SCHWARZ-CHRISTOFFEL MAPPING . . .
175
14.4.2 NUMERICAL APPROXIMATION TO A MAPPING ONTO A SMOOTH DOMAIN 179
APPENDIX TO CHAPTER 14: A PICTORIAL CATALOG OF CONFORMAL MAPS 181
15 TRANSFORM THEORY 195
15.0 INTRODUCTORY REMARKS 195
15.1 FOURIER SERIES 195
15.1.1 BASIC DEFINITIONS 195
15.1.2 A REMARK ON INTERVALS OF ARBITRARY LENGTH . . .. 196
15.1.3 CALCULATING FOURIER COEFFICIENTS 197
15.1.4 CALCULATING FOURIER COEFFICIENTS USING COMPLEX ANALYSIS 198
15.1.5 STEADY STATE HEAT DISTRIBUTION 199
15.1.6 THE DERIVATIVE AND FOURIER SERIES 201
15.2 THE FOURIER TRANSFORM 202
15.2.1 BASIC DEFINITIONS 202
15.2.2 SOME FOURIER TRANSFORM EXAMPLES THAT USE COMPLEX VARIABLES 203
IMAGE 13
XVIII CONTENTS
15.2.3 SOLVING A DIFFERENTIAL EQUATION USING THE FOURIER TRANSFORM 210
15.3 THE LAPLACE TRANSFORM 212
15.3.1 PROLOGUE 212
15.3.2 SOLVING A DIFFERENTIAL EQUATION USING THE LAPLACE TRANSFORM 213
15.4 THE Z-TRANSFORM 214
15.4.1 BASIC DEFINITIONS 214
15.4.2 POPULATION GROWTH BY MEANS OF THE Z-TRANSFORM 215
16 C O M P U T ER PACKAGES FOR S T U D Y I NG C O M P L EX VARIABLES 219
16.0 INTRODUCTORY REMARKS 219
16.1 THE SOFTWARE PACKAGES 219
16.1.1 THE SOFTWARE F(Z) 219
16.1.2 MATHEMATICA 221
16.1.3 MAPLE 227
16.1.4 MATLAB . 229
16.1.5 R I C C I 229
GLOSSARY OF T E R MS FROM C O M P L EX VARIABLE T H E O RY A ND ANALYSIS
231
LIST OF N O T A T I ON 269
TABLE OF LAPLACE TRANSFORMS 273
A G U I DE TO T HE L I T E R A T U RE 275
REFERENCES 279
I N D EX 283
|
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| author | Krantz, Steven G. 1951- |
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| building | Verbundindex |
| bvnumber | BV012769822 |
| callnumber-first | Q - Science |
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| callnumber-raw | QA331.7 |
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| dewey-full | 515/.9 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.9 |
| dewey-search | 515/.9 |
| dewey-sort | 3515 19 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012769822 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:33:21Z |
| institution | BVB |
| isbn | 3764340118 0817640118 |
| language | English |
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| physical | XXIV, 290 S. Ill., graph. Darst. |
| publishDate | 1999 |
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| publisher | Birkhäuser |
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| spelling | Krantz, Steven G. 1951- Verfasser (DE-588)130535907 aut Handbook of complex variables Steven G. Krantz Boston [u.a.] Birkhäuser 1999 XXIV, 290 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Funktionentheorie - Anwendung Komplexe Funktion Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008683756&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Krantz, Steven G. 1951- Handbook of complex variables Funktionentheorie - Anwendung Komplexe Funktion Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4018935-1 |
| title | Handbook of complex variables |
| title_auth | Handbook of complex variables |
| title_exact_search | Handbook of complex variables |
| title_full | Handbook of complex variables Steven G. Krantz |
| title_fullStr | Handbook of complex variables Steven G. Krantz |
| title_full_unstemmed | Handbook of complex variables Steven G. Krantz |
| title_short | Handbook of complex variables |
| title_sort | handbook of complex variables |
| topic | Funktionentheorie - Anwendung Komplexe Funktion Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Funktionentheorie - Anwendung Komplexe Funktion Funktionentheorie |
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