Introduction to the theory of complex functions:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2002
|
| Series: | Series in pure mathematics
25 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 300 S. graph. Darst. |
| ISBN: | 9812380477 |
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Record in the Search Index
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| adam_text | SERIES IN PURE MATHEMATICS - VOLUME 25 INTRODUCTION TO THE THEORY OF
COMPLEX FUNCTIONS JIAN-KE LU SHOU-GUO ZHONG WUHAN UNIVERSITY, CHINA
SHI-QIANG LIU NINGXIA UNIVERSITY, CHINA WORLD SCIENTIFIC NEW JERSEY *
LONDON * SINGAPORE * HONG KONG CONTENTS PREFACE V CHAPTER 1 COMPLEX
NUMBERS AND COMPLEX FUNCTIONS §1.1 COMPLEX NUMBERS 1 1.1.1 FIELD OF
COMPLEX NUMBERS 1 1.1.2 GEOMETRIC REPRESENTATION OF COMPLEX NUMBERS 3
1.1.3 STEREOGRAPHIC PROJECTION, COMPLEX SPHERE, POINT AT INFINITY AND
EXTENDED (COMPLEX) PLANE 7 EXERCISES 1.1 8 §1.2 FUNCTIONS OF A COMPLEX
VARIABLE 10 1.2.1 CONCEPT OF FUNCTIONS OF A COMPLEX VARIABLE 10 1.2.2
LIMIT AND CONTINUITY OF COMPLEX FUNCTIONS 11 1.2.3 CONCEPT OF HOMOTOPY
AND CONNECTEDNESS OF REGIONS 13 1.2.4 ARGUMENT FUNCTION 17 EXERCISES 1.2
23 §1.3 COMPLEX SEQUENCES AND COMPLEX SERIES 24 1.3.1 SEQUENCES AND
SERIES WITH COMPLEX TERMS 24 1.3.2 SEQUENCES AND SERIES OF COMPLEX
FUNCTIONS 26 EXERCISES 1.3 26 EXERCISES OF CHAPTER 1 27 CHAPTER 2
FUNDAMENTALS OF ANALYTIC FUNCTIONS §2.1 ANALYTIC FUNCTIONS 28 2.1.1
DERIVATIVE AND ITS GEOMETRIC MEANING 28 2.1.2 CONCEPT OF ANALYTIC
FUNCTIONS 32 EXERCISES 2.1 35 §2.2 SOME ELEMENTARY ANALYTIC FUNCTIONS 36
2.2.1 POLYNOMIALS AND RATIONAL FUNCTIONS 36 VIII INTRODUCTION TO THE
THEORY OF COMPLEX FUNCTIONS 2.2.2 EXPONENTIAL FUNCTIONS 36 2.2.3
TRIGONOMETRIC FUNCTIONS AND HYPERBOLIC FUNCTIONS 38 2.2.4 LOGARITHMIC
FUNCTIONS 40 2.2.5 POWER FUNCTIONS AND RADICAL FUNCTIONS 44 2.2.6
UNIFORMIZATION OF GENERAL MULTI-VALUED FUNCTIONS 47 2.2.7 LOGARITHM OF
RATIONAL FUNCTIONS 52 2.2.8 RADICALS OF RATIONAL FUNCTIONS 55 2.2.9
ANTI-TRIGONOMETRIC FUNCTIONS AND ANTI-HYPERBOLIC FUNCTIONS 58 EXERCISES
2.2 59 EXERCISES OF CHAPTER 2 61 CHAPTER 3 COMPLEX INTEGRALS §3.1 THE
CONCEPT OF COMPLEX INTEGRALS 63 3.1.1 DEFINITION OF COMPLEX INTEGRALS
AND THEIR CALCULATIONS 63 3.1.2 BASIC PROPERTIES OF COMPLEX INTEGRALS 67
EXERCISES 3.1 68 §3.2 FUNDAMENTAL THEOREMS 70 3.2.1 CAUCHY THEOREM ON
INTEGRALS 70 3.2.2 PRIMITIVE FUNCTIONS 76 EXERCISES 3.2 80 §3.3 BASIC
FORMULAS 81 3.3.1 CAUCHY INTEGRAL FORMULA 81 3.3.2 CAUCHY FORMULA OF
DIFFERENTIATION 83 3.3.3 CAUCHY INEQUALITY 86 3.3.4 MORERA THEOREM 86
EXERCISES 3.3 87 §3.4 IMPROPER COMPLEX INTEGRALS 88 3.4.1 DEFINITION OF
IMPROPER COMPLEX INTEGRALS 88 3.4.2 CAUCHY PRINCIPAL VALUE INTEGRALS 91
3.4.3 SINGULAR INTEGRALS OF HIGHER ORDER 95 EXERCISES 3.4 . 97 EXERCISES
OF CHAPTER 3 98 CONTENTS IX CHAPTER 4 THEORY OF SERIES FOR ANALYTIC
FUNCTIONS §4.1 GENERAL THEORY 100 4.1.1 INTEGRATION AND DIFFERENTIATION
TERM BY TERM FOR SERIES WITH COMPLEX FUNCTIONAL TERMS 100 4.1.2 POWER
SERIES AND ITS SUM FUNCTION 102 EXERCISES 4.1 104 §4.2 TAYLOR EXPANSION
AND UNIQUENESS THEOREM 105 4.2.1 TAYLOR EXPANSION OF ANALYTIC FUNCTIONS
105 4.2.2 UNIQUENESS OF ANALYTIC FUNCTIONS 113 4.2.3 MAXIMUM MODULUS
PRINCIPLE 115 EXERCISES 4.2 117 §4.3 LAURENT EXPANSION AND ISOLATED
SINGULARITIES 118 4.3.1 LAURENT EXPANSION OF ANALYTIC FUNCTIONS 118
4.3.2 METHODS FOR DETERMINING LAURENT EXPANSIONS 122 4.3.3 ISOLATED
SIGULARITIES OF ANALYTIC FUNCTIONS 126 4.3.4 ENTIRE FUNCTIONS AND
MEROMORPHIC FUNCTIONS 132 EXERCISES 4.3 135 EXERCISES OF CHAPTER 4 137
CHAPTER 5 THEORY OF RESIDUE §5.1 RESIDUE AND ITS EVALUATION 138 5.1.1
CONCEPT OF RESIDUE 138 5.1.2 RESIDUE AT THE POINT AT INFINITY 143 5.1.3
THE CASE OF BOUNDARY POINTS 145 EXERCISES 5.1 147 §5.2 RESIDUE THEOREM
AND ITS GENERALIZATION 148 5.2.1 RESIDUE THEOREM 148 5.2.2 EXTENDED
RESIDUE THEOREM 151 EXERCISES 5.2 154 §5.3 APPLICATIONS TO EVALUATION OF
INTEGRALS 155 5.3.1 APPLICATION OF SINGLE-VALUED ANALYTIC FUNCTIONS 156
5.3.2 APPLICATION OF MULTI-VALUED ANALYTIC FUNCTIONS 162 5.3.3
APPLICATION OF SINGULAR INTEGRALS OF HIGHER ORDER 169 X INTRODUCTION TO
THE THEORY OF COMPLEX FUNCTIONS EXERCISES 5.3 170 §5.4 ARGUMENT
PRINCIPLE AND ROUCHE THEOREM 170 5.4.1 ARGUMENT PRINCIPLE 171 5.4.2
ROUCHE THEOREM 173 EXERCISES 5.4 175 EXERCISES OF CHAPTER 5 175 CHAPTER
6 ANALYTIC EXTENSION §6.1 CONCEPT AND METHOD OF ANALYTIC EXTENSION 178
6.1.1 BASIC CONCEPT OF ANALYTIC EXTENSION 178 6.1.2 EXTENSION THROUGH
ARC 180 6.1.3 EXTENSION BY POWER SERIES 185 EXERCISES 6.1 190 §6.2
COMPLETE ANALYTIC FUNCTIONS AND MONODROME THEOREM 191 6.2.1 COMPLETE
ANALYTIC FUNCTIONS AND RIEMANN SURFACES 191 6.2.2 MONODROME THEOREM 193
EXERCISES 6.2 198 EXERCISES OF CHAPTER 6 198 CHAPTER 7 CONFORMAL MAPPING
§7.1 FRACTIONAL LINEAR MAPPING 200 7.1.1 CONFORMAL PROPERTY 201 7.1.2
GROUP OF MAPPINGS, FIXED POINTS 203 7.1.3 DETERMINATION OF A FLM,
INVARIANCE OF CROSS-RATIO 204 7.1.4 PRESERVATION OF CIRCLES 205 7.1.5
PRESERVATION OF SYMMETRIC POINTS 208 7.1.6 THREE SPECIAL FLMS 211
EXERCISES 7.1 215 §7.2 GENERAL THEORY OF CONFORMAL MAPPING 215 7.2.1
REGION PRESERVING FOR ANALYTIC FUNCTIONS AND PROPERTIES OF UNIVALENT
FUNCTIONS 216 7.2.2 RIEMANN MAPPING THEOREM, AND SCHWARZ LEMMA 219 7.2.3
BOUNDARY CORRESPONDENCE THEOREM 223 CONTENTS XI EXERCISES 7.2 225 §7.3
SOME ELEMENTARY FUNCTION MAPPINGS 225 7.3.1 MAPPINGS BY EXPONENTIAL OR
LOGARITHMIC FUNCTIONS 226 7.3.2 MAPPINGS BY POWER FUNCTIONS 227 7.3.3
JUKOVSKY FUNCTION 230 7.3.4 MAPPING BY THE COSINE FUNCTION 232 EXERCISES
7.3 234 §7.4 MISCELLANEOUS EXAMPLES 235 7.4.1 DETERMINE REGIONS BY GIVEN
FUNCTIONS 235 7.4.2 DETERMINE MAPPING FUNCTIONS BY GIVEN CORRESPONDING
REGIONS 236 EXERCISES 7.4 248 EXERCISES OF CHAPTER 7 249 CHAPTER 8
HARMONIC FUNCTIONS §8.1 CONCEPT OF HARMONIC FUNCTIONS AND THEIR
PROPERTIES 252 8.1.1 RELATIONSHIP BETWEEN HARMONIC FUNCTIONS AND
ANALYTIC FUNCTIONS 252 8.1.2 EXTREMUM PRINCIPLE 256 8.1.3 POISSON
FORMULA 257 EXERCISES 8.1 259 §8.2 DIRICHLET PROBLEMS . 260 8.2.1
GENERAL DIRICHLET PROBLEMS 260 8.2.2 PROPERTIES OF POISSON INTEGRAL 261
8.2.3 DIRICHLET PROBLEM ON A CIRCULAR REGION 264 8.2.4 DIRICHLET PROBLEM
ON THE UPPER HALF-PLANE 265 EXERCISES 8.2 266 §8.3 SCHWARZ-CHRISTOFFEL
FORMULA 267 8.3.1 GENERAL FORMULA 267 8.3.2 SOME EXAMPLES * 272
EXERCISES 8.3 276 EXERCISES OF CHAPTER 8 277 CHAPTER 9 ANALYTIC
FUNCTIONS APPLIED TO PLANAR FLOW §9.1 MEANING OF ANALYTIC FUNCITONS IN
FLUID MECHANICS 280 XII INTRODUCTION TO THE THEORY OF COMPLEX FUNCTIONS
9.1.1 COMPLEX FLOW 280 9.1.2 COMPLEX POTENTIAL 282 9.1.3 SOURCES AND
VORTICES 284 EXERCISES 9.1 287 §9.2 FLOW AROUND A CYLINDER 287 9.2.1
FLOW AROUND A CIRCULAR DISK 287 9.2.2 FLOW AROUND A GENERAL REGION 290
9.2.3 JUKOVSKY FORMULA OF LIFT-FORCE 292 EXERCISE 9.2 294 REFERENCES 295
INDEX 297
|
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| indexdate | 2024-07-09T19:13:28Z |
| institution | BVB |
| isbn | 9812380477 |
| language | English |
| lccn | 2003268755 |
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| physical | XII, 300 S. graph. Darst. |
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| publisher | World Scientific |
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| series | Series in pure mathematics |
| series2 | Series in pure mathematics |
| spelling | Lu, Jian-Ke Verfasser aut Introduction to the theory of complex functions Jian-Ke Lu ; Shou-Guo Zhong ; Shi-Qiang Liu Singapore [u.a.] World Scientific 2002 XII, 300 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Series in pure mathematics 25 Functions of complex variables Mehrere komplexe Variable (DE-588)4169285-8 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Funktion Mathematik (DE-588)4071510-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s DE-604 Funktion Mathematik (DE-588)4071510-3 s Mehrere komplexe Variable (DE-588)4169285-8 s Zhong, Shou-Guo Verfasser aut Liu, Shi-Qiang Verfasser aut Series in pure mathematics 25 (DE-604)BV000016845 25 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010300471&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lu, Jian-Ke Zhong, Shou-Guo Liu, Shi-Qiang Introduction to the theory of complex functions Series in pure mathematics Functions of complex variables Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktionentheorie (DE-588)4018935-1 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| subject_GND | (DE-588)4169285-8 (DE-588)4018935-1 (DE-588)4071510-3 |
| title | Introduction to the theory of complex functions |
| title_auth | Introduction to the theory of complex functions |
| title_exact_search | Introduction to the theory of complex functions |
| title_full | Introduction to the theory of complex functions Jian-Ke Lu ; Shou-Guo Zhong ; Shi-Qiang Liu |
| title_fullStr | Introduction to the theory of complex functions Jian-Ke Lu ; Shou-Guo Zhong ; Shi-Qiang Liu |
| title_full_unstemmed | Introduction to the theory of complex functions Jian-Ke Lu ; Shou-Guo Zhong ; Shi-Qiang Liu |
| title_short | Introduction to the theory of complex functions |
| title_sort | introduction to the theory of complex functions |
| topic | Functions of complex variables Mehrere komplexe Variable (DE-588)4169285-8 gnd Funktionentheorie (DE-588)4018935-1 gnd Funktion Mathematik (DE-588)4071510-3 gnd |
| topic_facet | Functions of complex variables Mehrere komplexe Variable Funktionentheorie Funktion Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010300471&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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