A history of complex dynamics: from Schröder to Fatou and Julia
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Braunschweig u.a.
Vieweg
1994
|
| Series: | Aspects of mathematics / E
24 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [143] - 162 |
| Physical Description: | VII, 165 S. graph. Darst. |
| ISBN: | 3528065206 |
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| 100 | 1 | |a Alexander, Daniel S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A history of complex dynamics |b from Schröder to Fatou and Julia |c Daniel S. Alexander |
| 264 | 1 | |a Braunschweig u.a. |b Vieweg |c 1994 | |
| 300 | |a VII, 165 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Aspects of mathematics / E |v 24 | |
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Record in the Search Index
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| adam_text | Contents
Preface 1
1 Schroder, Cayley and Newton s Method 3
1.1 Introduction 3
1.2 Schroder s Study of Iteration 4
1.3 Schroder s Fixed Point Theorem 6
1.4 A Generalization of Newton s Method 9
1.5 Schroder s Paper [1871] 11
1.6 Schroder and Functional Equations 12
1.7 Schroder and Newton s Method for the Quadratic 16
1.8 Arthur Cayley and Newton s Method 19
2 The Next Wave: Korkine and Farkas 23
2.1 Introduction 23
2.2 Analytic Iteration 24
2.3 Korkine and the Influence of Abel 28
2.4 Abel s Study of Functional Equations 29
2.5 Korkine s Solution to the Abel Equation 31
2.6 Farkas Solution to the Schroder Equation 34
3 Gabriel Koenigs 37
3.1 Gabriel Koenigs 37
3.2 Koenigs and Darboux 38
3.3 The Background to Koenigs Study of Iteration 40
3.4 Koenigs Study of Fixed Points 41
3.5 Koenigs Solution of the Schroder Equation 46
3.6 Koenigs and Functional Equations 47
3.7 Koenigs and the Global Study of Iteration 50
4 Iteration in the 1890 s: Grevy 53
4.1 A Brief Survey of Iteration in the 1890 s 53
4.2 Appell s Application of Koenigs Work to Hill s Differential Equation 54
4.3 Grevy and the Superattracting Case 55
vi CONTENTS
5 Iteration in the 1890 s: Leau 60
5.1 Basic Results in the | £ (0)| = 1 Case 60
5.2 Lemeray 64
5.3 Leau sWork 67
5.4 Leau s Anticipation of the Flower Theorem 68
5.5 Leau and Functional Equations 73
6 The Flower Theorem of Fatou and Julia 75
6.1 The Approaches of Fatou and Julia 75
6.2 Julia s Proof of the Flower Theorem 76
6.3 Fatou s Proof of the Flower Theorem 78
7 Fatou s 1906 Note 84
7.1 Introduction: Local Versus Global Studies of Iteration 84
7.2 The Lack of Set Theory in Koenigs Work 85
7.3 Fatou s Application of Set Theory 87
7.4 Pierre Fatou 89
7.5 Fatou s 1906 Note 90
7.6 Fatou and Functional Equations 94
8 Montel s Theory of Normal Families 96
8.1 Introduction 96
8.2 Paul Montel and Normal Families 98
8.3 The Influence of Ascoli and Arzela 99
8.4 Montel s Early Work 101
8.5 Montel s Study of Convergence Issues 103
8.6 An Important Result from [1912] 104
8.7 Applications of Montel s Theory to Picard Theory 105
9 The Contest 108
9.1 Overview 108
9.2 Biographical Sketches of Lattes and Julia 108
9.3 The 1918 Grand Prix 109
9.4 The Awarding of the Prize 113
10 Lattes and Ritt 117
10.1 Biographical Sketch of Ritt 117
10.2 The Approach of Lattes and Ritt 118
11 Fatou and Julia 124
11.1 Introduction to the Studies of Fatou and Julia 124
11.2 Iteration and the Theory of Normal Families 125
11.3 The Julia Set 126
11.4 Some Interesting Julia Sets 128
CONTENTS vii
11.5 Further Properties of the Julia Set 131
11.6 Iteration on the Fatou Set 134
11.7 A Limit on the Number of Attracting Orbits 136
11.8 The Number of Components of the Fatou Set 139
11.9 Newton s Method Again 140
Bibliography 143
Index 163
|
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| indexdate | 2024-07-09T17:19:25Z |
| institution | BVB |
| isbn | 3528065206 |
| language | English |
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| series2 | Aspects of mathematics / E |
| spelling | Alexander, Daniel S. Verfasser aut A history of complex dynamics from Schröder to Fatou and Julia Daniel S. Alexander Braunschweig u.a. Vieweg 1994 VII, 165 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Aspects of mathematics / E 24 Literaturverz. S. [143] - 162 Applications (Mathématiques) Cayley Jussieu Dynamique - Histoire ram Famille normale Jussieu Farkas Jussieu Fatou Jussieu Fonctions analytiques Fonctions, Théorie géométrique des Grévy Jussieu Histoire dynamique complexe Jussieu Itération (Mathématiques) Itération Jussieu Julia Jussieu Koenigs Jussieu Korkine Jussieu Lattès Jussieu Leau Jussieu Montel Jussieu Méthode Newton Jussieu Ritt Jussieu Schröder Jussieu Analytic functions Geometric function theory Iterative methods (Mathematics) Mappings (Mathematics) Geschichte (DE-588)4020517-4 gnd rswk-swf Iteration (DE-588)4123457-1 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Analytische Funktion (DE-588)4142348-3 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Iteration (DE-588)4123457-1 s Geschichte (DE-588)4020517-4 s DE-604 Analytische Funktion (DE-588)4142348-3 s Funktionentheorie (DE-588)4018935-1 s Dynamisches System (DE-588)4013396-5 s E Aspects of mathematics 24 (DE-604)BV000018737 24 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005546621&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Alexander, Daniel S. A history of complex dynamics from Schröder to Fatou and Julia Applications (Mathématiques) Cayley Jussieu Dynamique - Histoire ram Famille normale Jussieu Farkas Jussieu Fatou Jussieu Fonctions analytiques Fonctions, Théorie géométrique des Grévy Jussieu Histoire dynamique complexe Jussieu Itération (Mathématiques) Itération Jussieu Julia Jussieu Koenigs Jussieu Korkine Jussieu Lattès Jussieu Leau Jussieu Montel Jussieu Méthode Newton Jussieu Ritt Jussieu Schröder Jussieu Analytic functions Geometric function theory Iterative methods (Mathematics) Mappings (Mathematics) Geschichte (DE-588)4020517-4 gnd Iteration (DE-588)4123457-1 gnd Funktionentheorie (DE-588)4018935-1 gnd Analytische Funktion (DE-588)4142348-3 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4020517-4 (DE-588)4123457-1 (DE-588)4018935-1 (DE-588)4142348-3 (DE-588)4013396-5 |
| title | A history of complex dynamics from Schröder to Fatou and Julia |
| title_auth | A history of complex dynamics from Schröder to Fatou and Julia |
| title_exact_search | A history of complex dynamics from Schröder to Fatou and Julia |
| title_full | A history of complex dynamics from Schröder to Fatou and Julia Daniel S. Alexander |
| title_fullStr | A history of complex dynamics from Schröder to Fatou and Julia Daniel S. Alexander |
| title_full_unstemmed | A history of complex dynamics from Schröder to Fatou and Julia Daniel S. Alexander |
| title_short | A history of complex dynamics |
| title_sort | a history of complex dynamics from schroder to fatou and julia |
| title_sub | from Schröder to Fatou and Julia |
| topic | Applications (Mathématiques) Cayley Jussieu Dynamique - Histoire ram Famille normale Jussieu Farkas Jussieu Fatou Jussieu Fonctions analytiques Fonctions, Théorie géométrique des Grévy Jussieu Histoire dynamique complexe Jussieu Itération (Mathématiques) Itération Jussieu Julia Jussieu Koenigs Jussieu Korkine Jussieu Lattès Jussieu Leau Jussieu Montel Jussieu Méthode Newton Jussieu Ritt Jussieu Schröder Jussieu Analytic functions Geometric function theory Iterative methods (Mathematics) Mappings (Mathematics) Geschichte (DE-588)4020517-4 gnd Iteration (DE-588)4123457-1 gnd Funktionentheorie (DE-588)4018935-1 gnd Analytische Funktion (DE-588)4142348-3 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Applications (Mathématiques) Cayley Dynamique - Histoire Famille normale Farkas Fatou Fonctions analytiques Fonctions, Théorie géométrique des Grévy Histoire dynamique complexe Itération (Mathématiques) Itération Julia Koenigs Korkine Lattès Leau Montel Méthode Newton Ritt Schröder Analytic functions Geometric function theory Iterative methods (Mathematics) Mappings (Mathematics) Geschichte Iteration Funktionentheorie Analytische Funktion Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005546621&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018737 |
| work_keys_str_mv | AT alexanderdaniels ahistoryofcomplexdynamicsfromschrodertofatouandjulia |