Asymptotics and Borel summability:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2009
|
| Series: | Chapman & Hall/CRC monographs and surveys in pure and applied mathematics
141 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 250 S. graph. Darst. |
| ISBN: | 9781420070316 |
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MARC
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Record in the Search Index
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| adam_text | Titel: Asymptotics and borel summability
Autor: Costin, Ovidiu
Jahr: 2009
Contents
Introduction 1
1.1 Expansions and approximations ................ 1
1.1a Asymptotic expansions.................. 3
1.1b Functions asymptotic to an expansion, in the sense of
Poincare.......................... 3
1.1c Asymptotic power series................. 6
l.ld Operations with asymptotic power series........ 6
1.2 Formal and actual solutions................... 9
1.2a Limitations of representation of functions by expansions 10
1.2b Summation of a divergent series ............ 14
Review of some basic tools 19
2.1 The Phragmen-Lindelof theorem................ 19
2.2 Laplace and inverse Laplace transforms ............ 20
2.2a Inverse Laplace space convolution ........... 22
Classical asymptotics 25
3.1 Asymptotics of integrals: First results............. 25
3.1a Discussion: Laplace s method for solving ODEs of the
form ELofafc* + bk)y V =0.............. 26
3.2 Laplace, stationary phase, saddle point methods and Watson s
lemma .............................. 27
3.3 The Laplace method....................... 28
3.4 Watson s lemma ......................... 31
3.4a The Borel-Ritt lemma.................. 32
3.4b Laplace s method revisited: Reduction to Watson s
lemma........................... 34
3.5 Oscillatory integrals and the stationary phase method .... 36
3.5a Stationary phase method................ 39
3.5b Analytic integrands ................... 41
3.5c Examples......................... 42
3.6 Steepest descent method .................... 44
3.6a Further discussion of steepest descent lines...... 48
3.6b Reduction to Watson s lemma.............. 49
3.7 Application: Asymptotics of Taylor coefficients of analytic func-
tions ............................... 52
3.8 Banach spaces and the contractive mapping principle .... 55
3.8a Fixed points and vector valued analytic functions ... 57
3.8b Choice of the contractive map.............. 58
3.9 Examples............................. 59
3.9a Linear differential equations in Banach spaces..... 59
3.9b A Puiseux series for the asymptotics of the Gamma
function.......................... 59
3.9c The Gamma function .................. 61
3.9d Linear meromorphic differential equations. Regular and
irregular singularities................... 61
3.9e Spontaneous singularities: The Painleve s equation Pi 64
3.9f Discussion: The Painleve property........... 66
3.9g Irregular singularity of a nonlinear differential equation 67
3.9h Proving the asymptotic behavior of solutions of nonlin-
ear ODEs: An example ................. 68
3.9i Appendix: Some computer algebra calculations .... 69
3.10 Singular perturbations ..................... 70
3.10a Introduction to the WKB method ........... 70
3.10b Singularly perturbed Schrodinger equation: Setting and
heuristics......................... 71
3.10c Formal reexpansion and matching ........... 73
3.10d The equation in the inner region; matching subregions 74
3.10e Outer region: Rigorous analysis............. 74
3.10f Inner region: Rigorous analysis............. 77
3.10g Matching......................... 79
3.11 WKB on a PDE ......................... 79
Analyzable functions and transseries 81
4.1 Analytic function theory as a toy model of the theory of ana-
lyzable functions......................... 81
4.1a Formal asymptotic power series............. 83
4.2 Transseries ............................ 92
4.2a Remarks about the form of asymptotic expansions . . 92
4.2b Construction of transseries: A first sketch....... 92
4.2c Examples of transseries solution: A nonlinear ODE . . 96
4.3 Solving equations in terms of Laplace transforms....... 97
4.3a A second order ODE: The Painleve equation Pi ... . 102
4.4 Borel transform, Borel summation............... 103
4.4a The Borel transform B.................. 103
4.4b Definition of Borel summation and basic properties . . 104
4.4c Further properties of Borel summation......... 106
4.4d Stokes phenomena and Laplace transforms:
An example........................ 109
4.4e Nonlinear Stokes phenomena and formation of singular-
ities ............................ Ill
4.4f Limitations of classical Borel summation........ 112
4.5 Gevrey classes, least term truncation and Borel
summation ............................ 113
4.5a Connection between Gevrey asymptotics and Borel sum-
mation .......................... 115
4.6 Borel summation as analytic continuation........... 118
4.7 Notes on Borel summation ................... 119
4.7a Choice of critical time.................. 119
4.7b Discussion: Borel summation and differential and dif-
ference systems...................... 121
4.8 Borel transform of the solutions of an example ODE, (4.54) . 122
4.9 Appendix: Rigorous construction of transseries........ 122
4.9a Abstracting from §4.2b ................. 123
4.10 Logarithmic-free transseries................... 134
4.10a Inductive assumptions.................. 134
4.10b Passing from step N to step JV + 1........... 137
4.10c General logarithmic-free transseries........... 140
4.10d Ecalle s notation..................... 140
4.10e The space T of general transseries........... 142
Borel summability in differential equations 145
5.1 Convolutions revisited...................... 145
5.1a Spaces of sequences of functions ............ 147
5.2 Focusing spaces and algebras.................. 148
5.3 Example: Borel summation of the formal solutions to (4.54) 149
5.3a Borel summability of the asymptotic series solution . . 149
5.3b Borel summation of the transseries solution...... 150
5.3c Analytic structure along R+............... 152
5.4 General setting.......................... 154
5.5 Normalization procedures: An example ............ 154
5.6 Further assumptions and normalization ............ 156
5.6a Nonresonance....................... 156
5.6b The transseries solution of (5.51)............ 157
5.7 Overview of results ....................... 157
5.8 Further notation......................... 158
5.8a Regions in the p plane.................. 158
5.8b Ordering on Nn...................... 160
5.8c Analytic continuations between singularities...... 160
5.9 Analytic properties of Yk and resurgence........... 160
5.9a Summability of the transseries ............. 162
5.10 Outline of the proofs ...................... 164
5.10a Summability of the transseries in nonsingular directions:
A sketch.......................... 164
5.10b Higher terms of the transseries............. 166
5.10c Detailed proofs, for Re(cri) 0 and a 1 parameter
transseries......................... 167
5.10d Comments......................... 171
5.10e The convolution equation away from singular rays . . 172
5.10f Behavior of Y0(p) near p = 1.............. 176
5.10g General solution of (5.89) on [0,1 + e]......... 181
5.10h The solutions of (5.89) on [0,oo)............ 185
5.10i General L oc solution of the convolution equation . . . 187
5.10j Equations and properties of Yk and summation of the
transseries......................... 188
5.10k Analytic structure, resurgence, averaging ....... 193
5.11 Appendix............................. 198
5.11a AC(f * g) versus AC(f) * AC{g)............ 198
5.11b Derivation of the equations for the transseries for gen-
eral ODEs......................... 199
5.11c Appendix: Formal diagonalization........... 201
5.12 Appendix: The C*-algebra of staircase distributions, Vm u . 202
Asymptotic and transasymptotic matching; formation of sin-
gularities 211
6.0a Transseries and singularities: Discussion........ 212
6.1 Transseries reexpansion and singularities. Abel s equation . . 213
6.2 Determining the £ reexpansion in practice........... 215
6.3 Conditions for formation of singularities............ 216
6.4 Abel s equation, continued ................... 218
6.4a Singularities of Fq and proof of Lemma 6.35...... 220
6.5 General case ........................... 222
6.5a Notation and further assumptions ........... 223
6.5b The recursive system for the Fms............ 225
6.5c General results and properties of the functions Fm . . 226
6.6 Further examples ........................ 228
6.6a The Painleve equation Pi................ 228
6.6b The Painleve equation Pn................ 232
Other classes of problems 235
7.1 Difference equations....................... 235
7.1a Setting .......................... 235
7.1b Transseries for difference equations........... 236
7.1c Application: Extension of solutions yn of difference equa-
tions to the complex n plane.............. 236
7. Id Extension of the Painleve criterion to difference equa-
tions ............................ 237
7.2 PDEs ............................... 237
7.2a Example: Regularizing the heat equation....... 238
7.2b Higher order nonlinear svstems of evolution PDEs . . 239
8 Other important tools and developments 241
8.1 Resurgence, bridge equations, alien calculus, moulds..... 241
8.2 Multisummability ........................ 241
8.3 Hyperasymptotics ........................ 242
References 245
Index 249
|
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| discipline | Mathematik |
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| id | DE-604.BV035910176 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:07:14Z |
| institution | BVB |
| isbn | 9781420070316 |
| language | English |
| lccn | 2008034107 |
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| physical | XIII, 250 S. graph. Darst. |
| publishDate | 2009 |
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| publisher | CRC Press |
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| series | Chapman & Hall/CRC monographs and surveys in pure and applied mathematics |
| series2 | Chapman & Hall/CRC monographs and surveys in pure and applied mathematics |
| spelling | Costin, Ovidiu 1960- Verfasser (DE-588)137496893 aut Asymptotics and Borel summability Ovidiu Costin Boca Raton [u.a.] CRC Press 2009 XIII, 250 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC monographs and surveys in pure and applied mathematics 141 Differential equations / Asymptotic theory Summability theory Borel-Summierungsverfahren (DE-588)4146328-6 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Asymptotik (DE-588)4126634-1 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Asymptotik (DE-588)4126634-1 s Borel-Summierungsverfahren (DE-588)4146328-6 s DE-604 Chapman & Hall/CRC monographs and surveys in pure and applied mathematics 141 (DE-604)BV013350872 141 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018767467&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Costin, Ovidiu 1960- Asymptotics and Borel summability Chapman & Hall/CRC monographs and surveys in pure and applied mathematics Differential equations / Asymptotic theory Summability theory Borel-Summierungsverfahren (DE-588)4146328-6 gnd Differentialgleichung (DE-588)4012249-9 gnd Asymptotik (DE-588)4126634-1 gnd |
| subject_GND | (DE-588)4146328-6 (DE-588)4012249-9 (DE-588)4126634-1 |
| title | Asymptotics and Borel summability |
| title_auth | Asymptotics and Borel summability |
| title_exact_search | Asymptotics and Borel summability |
| title_full | Asymptotics and Borel summability Ovidiu Costin |
| title_fullStr | Asymptotics and Borel summability Ovidiu Costin |
| title_full_unstemmed | Asymptotics and Borel summability Ovidiu Costin |
| title_short | Asymptotics and Borel summability |
| title_sort | asymptotics and borel summability |
| topic | Differential equations / Asymptotic theory Summability theory Borel-Summierungsverfahren (DE-588)4146328-6 gnd Differentialgleichung (DE-588)4012249-9 gnd Asymptotik (DE-588)4126634-1 gnd |
| topic_facet | Differential equations / Asymptotic theory Summability theory Borel-Summierungsverfahren Differentialgleichung Asymptotik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018767467&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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| work_keys_str_mv | AT costinovidiu asymptoticsandborelsummability |