Divergent series, summability and resurgence III: resurgent methods and the first painlevé equation
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
[Cham]
Springer
[2016]
|
| Series: | Lecture Notes in Mathematics
2155 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxii, 230 Seiten Diagramme |
| ISBN: | 9783319289991 |
| ISSN: | 0075-8434 |
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MARC
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| 245 | 1 | 0 | |a Divergent series, summability and resurgence III |b resurgent methods and the first painlevé equation |c Eric Delabaere |
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Record in the Search Index
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| adam_text | Eric Delabaere
Divergent Series,
Summability and
Resurgence III
Resurgent Methods and the First
Painleve Equation
Springer
Contents
Avant-Propos vii
Preface to the Three Volumes ix
Preface to this Volume xvii
1 Some Elements about Ordinary Differential Equations 1
1 1 Ordinary Differential Equations in the Complex Domain 1
111 The Fundamental Existence Theorem 1
112 Some Usual Terminology 2
113 Algebraic Differential Equations 3
1 2 Solutions of Ordinary Differential Equations and Singularities 3
121 Notation 3
122 Problem 4
123 Linear Differential Equations 4
124 Nonlinear Differential Equations 5
1 3 The Painleve Program, Painleve Property and Painleve Equations 8
131 Ordinary Differential Equations of Order One 9
132 Ordinary Differential Equations of Order Two, Painleve
Equations 10
133 Painleve Equations and Related Topics 11
References 12
2 The First Painleve Equation 15
2 1 The First Painlevd Equation 15
2 2 Painleve Property for the First Painleve Equation 15
2 3 First Painleve Equation and Isomonodromic Deformations Condition 16
2 4 Lax Formalism 17
2 5 Symmetries 18
2 6 Asymptotics at Infinity 18
261 Dominant Balance Principle 19
xx Contents;
262 Maximal Balance, Elliptic Function-type Behavior 20 ,
263 Submaximal Dominant Balances, Truncated Solutions 21
2 7 First Painleve Equation and Physical Models 22,
Appendix 23
References 30
3 Tritruncated Solutions For The First Painleve Equation 33
3 1 Normalization and Formal Series Solution 33
311 Normalization, Prepared Form 33
312 Formal Series Solution 34
313 Towards Truncated Solutions 35
32A Reminder 37
3 3 Formal Series Solution and Borel-Laplace Summability 39
331 Formal Series Solution and Borel-Laplace Summability: a
Perturbative Approach 40
332 Formal Series Solution and Borel-Laplace Summability:
Second Approach 48
3 4 First Painleve Equation and Tritruncated Solutions 54
341 Reminder 54
342 Formal Series Solution and Borel-Laplace Summation 56
343 Tritruncated Solutions 61
Exercices 64
References 67
4 A Step Beyond Borel-Laplace Summability 69
4 1 Introduction 69
4 2 Resurgent Functions and Riemann Surface 71
421 Notation 71
422 The Riemann Surface of Z-Resurgent Functions 72
423 Riemann Surface and Sheets 74
424 Nearby Domains 77
425 Geodesics 80
4 3 Shortest Symmetric (Z,p)-Homotopy 81
431 Symmetric Z-Homotopy 81
432 Shortest Symmetric (Z,p)-Homotopy 82
4 4 Convolution Product and Related Properties 85
441A New Convolution Algebra 85
442 Convolution Space and Uniform Norm 87
443 An Extended Grönwall-like Lemma 89
4 5 Application to the First Painleve Equation 91
451A Step Beyond Borel-Laplace Summability 91
452 Concluding Remarks 93
4 6 Some Supplements 95
4 7 Comments 97
References 97
Contents
xxi
5
Transseries And Formal Integral For The First Painleve Equation 99
5 1 Introduction 99
5 2 Formal Integral: Setting 101
521 Notation 101
522 Setting 102
523 Formal Integral, General Considerations 107
5 3 First Painlevd Equation and Transseries Solutions 110
531 Transseries Solution - Statement Ill
532 Transseries Solution - Proof 112
5 4 Formal Integral for the First Painleve Equation 116
541 Notation and Preliminary Results 116
542 Painlevä I: Formal Integral 119
5 5 Comments 126
References 127
6 Truncated Solutions For The First Painleve Equation 129
6 1 Borel-Laplace Summability of the k-th Series and Beyond 129
611 Preliminary Results 129
612 The e;-th Series 131
613 The £e,-th Series 133
614 The Other k-th Series 136
6 2 Borel-Laplace Summability of the Transseries 136
621A Useful Supplement 136
622 Borel-Laplace Summability of the Transseries 138
623 Remarks 141
624 Considerations on the Domain 141
6 3 Summability of the Formal Integral 143
6 4 Truncated Solutions for the First Painleve Equation 143
6 5 Comments 145
References 146
7 Supplements To Resurgence Theory 147
7 1 Introduction 147
7 2 Sectorial Germs 149
721 Sectors 149
722 Sectorial Germs 149
7 3 Microfunctions 150
731 Microfunctions - Definitions 151
732 Microfunctions and Convolution Product 152
7 4 Space of Singularities 158
741 Singularities 158
742 Convolution Product of Singularities 160
7 5 Formal Laplace Transform, Formal Borel Transform 161
751 Formal Laplace Transform for Microfunctions 161
752 Formal Borel transform for asymptotic classes 164
xxii Contents
753 Formal Laplace Transform for Singularities and Back to
Convolution Product 166
7 6 Laplace Transforms 171
761 Laplace Transforms 171
762 Singularities and Laplace Transform 173
7 7 Spaces of Resurgent Functions 175
771 Preliminaries 175
772 Resurgent Functions 178
7 8 Alien Operators 183
781 Alien Operators Associated with a Triple 184
782 Composition of Alien Operators 189
783 Alien Derivations 196
7 9 Ramified Resurgent Functions 201
791 Simple and Simply Ramified Resurgent Functions 202
792 Ramified Resurgent Functions 203
7 10 Comments 206
References 206
8 Resurgent Structure For The First Painleve Equation 209
8 1 The Main Theorem 209
811 Reminder 209
812 The Main Theorem 210
8 2 Stokes Phenomenon 213
8 3 The Alien Derivatives for the Seen Singularities 214
831 Preparations 215
832 Alien Derivations, First Approach 217
833 Alien Derivations, Second Approach 218
834A Step Further 221
8 4 The Bridge Equation and Proof of the Main Theorem 222
8 5 Comments 224
References 225
Index
|
| any_adam_object | 1 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515.24 |
| dewey-search | 515.24 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-10T07:32:16Z |
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| isbn | 9783319289991 |
| issn | 0075-8434 |
| language | English |
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| spelling | Delabaere, Eric Verfasser (DE-588)1024385841 aut Divergent series, summability and resurgence III resurgent methods and the first painlevé equation Eric Delabaere [Cham] Springer [2016] xxii, 230 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Lecture Notes in Mathematics 2155 0075-8434 Mathematics Functions of complex variables Differential equations Sequences (Mathematics) Special functions Sequences, Series, Summability Ordinary Differential Equations Functions of a Complex Variable Special Functions Mathematik Painlevé-Gleichung (DE-588)4123902-7 gnd rswk-swf Painlevé-Gleichung (DE-588)4123902-7 s DE-604 Erscheint auch als Online-Ausgabe 978-3-319-29000-3 Lecture Notes in Mathematics 2155 (DE-604)BV000676446 2155 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029091206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Delabaere, Eric Divergent series, summability and resurgence III resurgent methods and the first painlevé equation Lecture Notes in Mathematics Mathematics Functions of complex variables Differential equations Sequences (Mathematics) Special functions Sequences, Series, Summability Ordinary Differential Equations Functions of a Complex Variable Special Functions Mathematik Painlevé-Gleichung (DE-588)4123902-7 gnd |
| subject_GND | (DE-588)4123902-7 |
| title | Divergent series, summability and resurgence III resurgent methods and the first painlevé equation |
| title_auth | Divergent series, summability and resurgence III resurgent methods and the first painlevé equation |
| title_exact_search | Divergent series, summability and resurgence III resurgent methods and the first painlevé equation |
| title_full | Divergent series, summability and resurgence III resurgent methods and the first painlevé equation Eric Delabaere |
| title_fullStr | Divergent series, summability and resurgence III resurgent methods and the first painlevé equation Eric Delabaere |
| title_full_unstemmed | Divergent series, summability and resurgence III resurgent methods and the first painlevé equation Eric Delabaere |
| title_short | Divergent series, summability and resurgence III |
| title_sort | divergent series summability and resurgence iii resurgent methods and the first painleve equation |
| title_sub | resurgent methods and the first painlevé equation |
| topic | Mathematics Functions of complex variables Differential equations Sequences (Mathematics) Special functions Sequences, Series, Summability Ordinary Differential Equations Functions of a Complex Variable Special Functions Mathematik Painlevé-Gleichung (DE-588)4123902-7 gnd |
| topic_facet | Mathematics Functions of complex variables Differential equations Sequences (Mathematics) Special functions Sequences, Series, Summability Ordinary Differential Equations Functions of a Complex Variable Special Functions Mathematik Painlevé-Gleichung |
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