Complex function theory:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
[2007]
|
| Ausgabe: | Second edition |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016410987&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xii, 163 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821844281 0821844288 |
Internformat
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Datensatz im Suchindex
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|---|---|
| adam_text | COMPLEX FUNCTION THEORY SECOND EDITION DONALD SARASON AMS AMERICAN
MATHEMATICAL SOCIETY CONTENTS PREFACE TO THE SECOND EDITION IX PREFACE
TO THE FIRST EDITION XI CHAPTER I. COMPLEX NUMBERS 1 §1.1. DEFINITION OF
C 2 §1.2. FIELD AXIOMS 2 §1.3. EMBEDDING OF R IN C. THE IMAGINARY UNIT 3
§1.4. GEOMETRIC REPRESENTATION 3 §1.5. TRIANGLE INEQUALITY 4 §1.6.
PARALLELOGRAM EQUALITY 5 §1.7. PLANE GEOMETRY VIA COMPLEX NUMBERS 5
§1.8. C AS A METRIC SPACE- 6 §1.9. POLAR FORM 6 §1.10. DE MOIVRE S
FORMULA 7 §1.11. ROOTS * 8 §1.12. STEREOGRAPHIC PROJECTION 9 §1.13.
SPHERICAL METRIC 10 §1.14. EXTENDED COMPLEX PLANE 11 CHAPTER II. COMPLEX
DIFFERENTIATION 13 §11.1. DEFINITION OF THE DERIVATIVE 13 §11.2.
RESTATEMENT IN TERMS OF LINEAR APPROXIMATION 14 §11.3. IMMEDIATE
CONSEQUENCES 14 III IV CONTENTS §11.4. POLYNOMIALS AND RATIONAL
FUNCTIONS 15 §11.5. COMPARISON BETWEEN DIFFERENTIABILITY IN THE REAL AND
COMPLEX SENSES 15 §11.6. CAUCHY-RIEMANN EQUATIONS 16 §11.7. SUFFICIENT
CONDITION FOR DIFFERENTIABILITY 17 §11.8. HOLOMORPHIC FUNCTIONS 17
§11.9. COMPLEX PARTIAL DIFFERENTIAL OPERATORS 18 §11.10. PICTURING A
HOLOMORPHIC FUNCTION 19 §11.11. CURVES IN C 20 §11.12. CONFORMALITY 21
§11.13. CONFORMAL IMPLIES HOLOMORPHIC 22 §11.14. HARMONIC FUNCTIONS 23
§11.15. HOLOMORPHIC IMPLIES HARMONIC 24 §11.16. HARMONIC CONJUGATES 24
CHAPTER III. LINEAR-FRACTIONAL TRANSFORMATIONS 27 §111.1. COMPLEX
PROJECTIVE SPACE 27 §111.2. LINEAR-FRACTIONAL TRANSFORMATIONS 28 §111.3.
CONFORMALITY 29 §111.4. FIXED POINTS 29 §111.5. THREE-FOLD TRANSITIVITY
29 §111.6. FACTORIZATION 30 §111.7. CLIRCLES 31 §111.8. PRESERVATION OF
CLIRCLES 31 §111.9. ANALYZING A LINEAR-FRACTIONAL TRANSFORMATION*AN
EXAMPLE 32 CHAPTER IV. ELEMENTARY FUNCTIONS 35 §IV.L. DEFINITION OF E Z
35 §IV.2. LAW OF EXPONENTS 36 §IV.3. E Z IS HOLOMORPHIC 36 §IV.4.
PERIODICITY 37 §IV.5. E Z AS A MAP 37 §IV.6. HYPERBOLIC FUNCTIONS 38
§IV.7. ZEROS OF COSHZ AND SINHZ. 38 §IV.8. TRIGONOMETRIC FUNCTIONS 39
§IV.9. LOGARITHMS 40 §IV.1O. BRANCHES OF ARG Z AND LOG Z. 40 CONTENTS
V . §IV.LL. LOG Z AS A HOLOMORPHIC FUNCTION 41 §IV.12. LOGARITHMS OF
HOLOMORPHIC FUNCTIONS 42 §IV.13. ROOTS 43 §IV.14. INVERSES OF
HOLOMORPHIC FUNCTIONS - 43 §IV.15. INVERSE TRIGONOMETRIC FUNCTIONS 44
§IV.16. POWERS 45 §IV.17. ANALYTIC CONTINUATION AND RIEMANN SURFACES 45
CHAPTER V. POWER SERIES 49 §V.L. INFINITE SERIES 49 §V.2. NECESSARY
CONDITION FOR CONVERGENCE 49 §V.3. GEOMETRIC SERIES 50 §V.4. TRIANGLE
INEQUALITY FOR SERIES 50 §V.5. ABSOLUTE CONVERGENCE 50 §V.6. SEQUENCES
OF FUNCTIONS 51 §V.7. SERIES OF FUNCTIONS 51 §V.8, POWER SERIES 52 §V.9.
REGION OF CONVERGENCE 53 §V.1O. RADIUS OF CONVERGENCE 54 §V.LL. LIMITS
SUPERIOR 54 §V.12. CAUCHY-HADAMARD THEOREM 55 §V.13. RATIO TEST 56
§V.14. EXAMPLES . * 57 §V.15. DIFFERENTIATION OF POWER SERIES . 58
§V.16. EXAMPLES - / 60 §V.17. CAUCHY PRODUCT . 61 §V.18. DIVISION OF
POWER SERIES 63 CHAPTER VI. COMPLEX INTEGRATION 65 §VI.L. RIEMANN
INTEGRAL FOR COMPLEX-VALUED FUNCTIONS 65 §VI.2. FUNDAMENTAL THEOREM OF
CALCULUS 66 §VI.3. TRIANGLE INEQUALITY FOR INTEGRATION : 66 §VI.4. ARC
LENGTH 67 §VI.5. THE COMPLEX INTEGRAL 67 §VI.6. INTEGRAL OF A DERIVATIVE
68 §VI.7. AN EXAMPLE 68 VI CONTENTS §VI.8. REPARAMETRIZATION 69 §VI.9.
THE REVERSE OF A CURVE 70 §VI.1O. ESTIMATE OF THE INTEGRAL 71 §VI.LL.
INTEGRAL OF A LIMIT 71 §VI.12. AN EXAMPLE 71 CHAPTER VII. CORE VERSIONS
OF CAUCHY S THEOREM, AND CONSEQUENCES 75 §VII.L. CAUCHY S THEOREM FOR A
TRIANGLE 75 §VII.2. CAUCHY S THEOREM FOR A CONVEX REGION 78 §VII.3.
EXISTENCE OF A PRIMITIVE 78 §VII.4. MORE DEFINITE INTEGRALS 79 §VII.5.
CAUCHY S FORMULA FOR A CIRCLE 79 §VII.6. MEAN VALUE PROPERTY 81 §VII.7.
CAUCHY INTEGRALS 82 §VII.8. IMPLICATIONS FOR HOLOMORPHIC FUNCTIONS 83
§VII.9. CAUCHY PRODUCT 84 §VII. 10. CONVERSE OF GOURSAT S LEMMA 85
§VII.LL. LIOUVILLE S THEOREM 86 §VII.12. FUNDAMENTAL THEOREM OF ALGEBRA
86 §VII.13. ZEROS OF HOLOMORPHIC FUNCTIONS 87 §VII. 14. THE IDENTITY
THEOREM 89 §VII. 15. WEIERSTRASS CONVERGENCE THEOREM 89 §VII. 16.
MAXIMUM MODULUS PRINCIPLE 90 §VII.17. SCHWARZ S LEMMA V 91 §VIL18.
EXISTENCE OF HARMONIC CONJUGATES 93 §VII. 19. INFINITE DIFFERENTIABILITY
OF HARMONIC FUNCTIONS 94 §VII.20. MEAN VALUE PROPERTY FOR HARMONIC
FUNCTIONS 94 §VII.21. IDENTITY THEOREM FOR HARMONIC FUNCTIONS 94
§VII.22. MAXIMUM PRINCIPLE FOR HARMONIC FUNCTIONS 95 §VII.23. HARMONIC
FUNCTIONS IN HIGHER DIMENSIONS 95 CHAPTER VIII. LAURENT SERIES AND
ISOLATED SINGULARITIES 97 §VIII.L. SIMPLE EXAMPLES 97 §VIIL2. LAURENT
SERIES 98 §VIIL3. CAUCHY INTEGRAL NEAR OO 99 §VIIL4. CAUCHY S THEOREM
FOR TWO CONCENTRIC CIRCLES 100 CONTENTS VII §VIII.5. CAUCHY S FORMULA
FOR AN ANNULUS 101 §VIII.6. EXISTENCE OF LAURENT SERIES REPRESENTATIONS
101 §VIII.7. ISOLATED SINGULARITIES 102 §VIII.8. CRITERION FOR A
REMOVABLE SINGULARITY 105 §VIII.9. CRITERION FOR A POLE 105 §VIII.10.
CASORATI-WEIERSTRASS THEOREM 106 §VIII.LL. PICARD S THEOREM 106 §VIIL12.
RESIDUES 106 CHAPTER IX. CAUCHY S THEOREM 109 §IX.L. CONTINUOUS
LOGARITHMS 109 §IX.2. PIECEWISE C 1 CASE 110 §IX.3. INCREMENTS IN THE
LOGARITHM AND ARGUMENT ALONG A CURVE 110 §IX.4. WINDING NUMBER 111
§IX.5. CASE OF A PIECEWISE-C 1 CURVE 111 §LX.6. CONTOURS 113 §IX.7.
WINDING NUMBERS OF CONTOURS 114 §IX.8. SEPARATION LEMMA 115 §IX.9.
ADDENDUM TO THE SEPARATION LEMMA 117 §IX.10. CAUCHY S THEOREM 118
§IX.LL. HOMOTOPY 119 §IX.12. CONTINUOUS LOGARITHMS*2-D VERSION 119
§IX.13. HOMOTOPY AND WINDING NUMBERS 120 §IX.14. HOMOTOPY VERSION OF
CAUCHY S THEOREM - 121 §IX.15. RUNGE S APPROXIMATION THEOREM F 121
§IX.16. SECOND PROOF OF CAUCHY S THEOREM 122 §IX.17. SHARPENED FORM OF
RUNGE S THEOREM 123 CHAPTER X. FURTHER DEVELOPMENT OF BASIC COMPLEX
FUNCTION 125 §X.L. SIMPLY CONNECTED DOMAINS 125 §X.2. WINDING NUMBER
CRITERION 126 §X.3. CAUCHY S THEOREM FOR SIMPLY CONNECTED DOMAINS 126
§X.4. EXISTENCE OF PRIMITIVES 127 §X.5. EXISTENCE OF LOGARITHMS 127
§X.6. EXISTENCE OF HARMONIC CONJUGATES 128 §X.7. SIMPLE CONNECTIVITY AND
HOMOTOPY 128 VIII CONTENTS §X;8. THE RESIDUE THEOREM 129 §X.9. CAUCHY S
FORMULA 130 §X.1O. MORE DEFINITE INTEGRALS 130 §X.LL. THE ARGUMENT
PRINCIPLE 137 §X.12. ROUCHE S THEOREM 138 §X.13. THE LOCAL MAPPING
THEOREM 140 §X.14. CONSEQUENCES OF THE LOCAL MAPPING THEOREM 140 §X.15.
INVERSES 141 §X.16. CONFORMAL EQUIVALENCE 141 §X.17. THE RIEMANN MAPPING
THEOREM 142 §X.18. AN EXTREMAL PROPERTY OF RIEMANN MAPS 143 §X.19.
STIELTJES-OSGOOD THEOREM 144 §X.2O. PROOF OF THE RIEMANN MAPPING THEOREM
146 §X.21. SIMPLE CONNECTIVITY AGAIN 148 APPENDIX 1. SUFFICIENT
CONDITION FOR DIFFERENTIABILITY 151 APPENDIX 2. TWO INSTANCES OF THE
CHAIN RULE 153 APPENDIX 3. GROUPS, AND LINEAR-FRACTIONAL TRANSFORMATIONS
155 APPENDIX 4. DIFFERENTIATION UNDER THE INTEGRAL SIGN 157 REFERENCES
159 INDEX 161
|
| any_adam_object | 1 |
| author | Sarason, Donald 1933- |
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| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.9 |
| dewey-sort | 3515 19 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
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| genre_facet | Einführung |
| id | DE-604.BV023225155 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:11:04Z |
| institution | BVB |
| isbn | 9780821844281 0821844288 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016410987 |
| oclc_num | 173368750 |
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| owner | DE-20 DE-703 DE-11 DE-188 DE-19 DE-BY-UBM |
| owner_facet | DE-20 DE-703 DE-11 DE-188 DE-19 DE-BY-UBM |
| physical | xii, 163 Seiten Illustrationen, Diagramme |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | American Mathematical Society |
| record_format | marc |
| spellingShingle | Sarason, Donald 1933- Complex function theory Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4151278-9 |
| title | Complex function theory |
| title_auth | Complex function theory |
| title_exact_search | Complex function theory |
| title_full | Complex function theory Donald Sarason |
| title_fullStr | Complex function theory Donald Sarason |
| title_full_unstemmed | Complex function theory Donald Sarason |
| title_short | Complex function theory |
| title_sort | complex function theory |
| topic | Functions of complex variables Funktionentheorie (DE-588)4018935-1 gnd |
| topic_facet | Functions of complex variables Funktionentheorie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016410987&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT sarasondonald complexfunctiontheory |