Divergent Series, Summability and Resurgence I: Monodromy and Resurgence PDF
Preview Divergent Series, Summability and Resurgence I: Monodromy and Resurgence
Lecture Notes in Mathematics 2153 Claude Mitschi David Sauzin Divergent Series, Summability and Resurgence I Monodromy and Resurgence Lecture Notes in Mathematics 2153 Editors-in-Chief: J.-M.Morel,Cachan B.Teissier,Paris AdvisoryBoard: CamilloDeLellis,Zürich MariodiBernardo,Bristol AlessioFigalli,Austin DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,ParisandNew York CatharinaStroppel,Bonn AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Claude Mitschi • David Sauzin Divergent Series, Summability and Resurgence I Monodromy and Resurgence Claude Mitschi David Sauzin Inst. de Recherche Mathématique Avancée CNRS UMR 8028 -- IMCCE Université de Strasbourg et CNRS Observatoire de Paris Strasbourg, France Paris, France ISSN 0075-8434 ISSN1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-319-28735-5 ISBN 978-3-319-28736-2 (eBook) DOI 10.1007/978-3-319-28736-2 LibraryofCongressControlNumber:2016940058 Mathem ati csSubjec tClassifi cation(20 10):34M30, 30E15, 30B40, 34M03, 34M40, 37F10, 34M35 © Springer International Publishing Switzerland 2016 This work is subject to copyright. 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Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland a` lame´moired’AndreyBolibrukh,C.M. a` Lili,D.S. Avant-Propos Le sujet principal traite´ dans la se´rie de volumes Divergent Series, Summability and Resurgence est la the´orie des de´veloppements asymptotiques et des se´ries di- vergentes applique´e aux e´quations diffe´rentielles ordinaires (EDO) et a` certaines e´quationsauxdiffe´rencesdanslechampcomplexe. Lese´quationsdiffe´rentiellesdanslechampcomplexe,etdanslecadreholomor- phe,sontunsujettre`sancien.Lathe´orieae´te´ tre`sactivedansladeuxie`memoitie´ duXIX-e`mesie`cle.Encequiconcernelese´quationsline´aires,lesmathe´maticiens de cette e´poque les ont subdivise´es en deux classes. Pour la premie`re, celle des e´quationsa` pointssinguliersre´guliers(oudeFuchs),ge´ne´ralisantlese´quationshy- perge´ome´triques d’Euler et de Gauss, ils ont enregistre´ “des succe`s aussi de´cisifs quefaciles”commel’e´crivaitRene´Garnieren1919.Enrevanche,pourlaseconde, celledese´quationsditesa` pointssinguliersirre´guliers,commel’e´crivaitaussiGar- nier,“leurseffortsrestentimpuissantsa` e´difieraucunethe´oriege´ne´rale”.Laraison centrale de ce vif contraste est que toute se´rie entie`re apparaissant dans l’e´criture d’unesolutiond’unee´quationdiffe´rentielledeFuchsestautomatiquementconver- gentetandisquepourlese´quationsirre´gulie`rescesse´riessontge´ne´riquementdiver- gentes et que l’on ne savait qu’en faire. La situation a commence´ a` changer graˆce a` un travail magistral de Henri Poincare´ entrepris juste apre`s sa the`se, dans lequel il “donne un sens” aux solutions divergentes des EDO line´aires irre´gulie`res en in- troduisant un outil nouveau, et qui e´tait appele´ a` un grand avenir, la the´orie des de´veloppementsasymptotiques.Ilaensuiteutilise´cetoutilpourdonnerunsensaux se´ries divergentes de la me´canique ce´leste, et remporte´ de tels succe`s que presque toutlemondeaoublie´ l’originedel’histoire,c’est-a`-direlesEDO!Lestravauxde Poincare´ont(unpeu...)remisa`l’honneurl’e´tudedesse´riesdivergentes,abandonne´e parlesmathe´maticiensapre`sCauchy.L’Acade´miedesSciencesasoumiscesujetau concoursen1899,cequifuta`l’origined’untravailimportantd’E´mileBorel.Celui- ciestlasourcedenombredestechniquesutilise´esdansDivergentSeries,Summabil- ityandResurgence.PourrevenirauxEDOirre´gulie`res,lesujetafaitl’objetdenom- breuxetimportantstravauxdeG.D.BirkhoffetR.Garnierdurantlepremierquart duXX-e`mesie`cle.Onretrouveraicidenombreuxprolongementsdesme´thodesde Birkhoff. Apre`s 1940, le sujet a e´trangement presque disparu, la the´orie e´tant, je vii viii Avant-Propos nesaistroppourquoi,conside´re´ecommeacheve´e,toutcommecelledese´quations de Fuchs. Ces dernie`res ont re´e´merge´ au de´but des anne´es 1970, avec les travaux deRaymondGe´rard,puisunlivredePierreDeligne.Lese´quationsirre´gulie`resont suiviavecdestravauxdel’e´coleallemandeetsurtoutdel’e´colefranc¸aise.Denom- breuses techniques comple`tement nouvelles ont e´te´ introduites (de´veloppements asymptotiques Gevrey, k-sommabilite´, multisommabilite´, fonctions re´surgentes...) permettant en particulier une vaste ge´ne´ralisation du phe´nome`ne de Stokes et sa miseenrelationaveclathe´oriedeGaloisdiffe´rentielleetleproble`medeRiemann- Hilbert ge´ne´ralise´. Tout ceci a depuis rec¸u de tre`s nombreuses applications dans des domaines tre`s varie´s, allant de l’inte´grabilite´ des syste`mes hamiltoniens aux proble`mes de points tournants pour les EDO singulie`rement perturbe´es ou a` divers proble`mes de modules. On en trouvera certaines dans Divergent Series, Summa- bilityandResurgence,commel’e´tudere´surgentedesgermesdediffe´omorphismes analytiques du plan complexe tangents a` l’identite´ ou celle de l’EDO non-line´aire Painleve´ I. Lesujetrestaitaujourd’huidifficiled’acce`s,lelecteurnedisposantpas,misa`part les articles originaux, de pre´sentation accessible couvrant tous les aspects. Ainsi Divergent Series, Summability and Resurgence comble une lacune. Ces volumes pre´sententunlargepanoramadesrechercheslesplusre´centessurunvastedomaine classique et passionnant, en pleine renaissance, on peut meˆme dire en pleine ex- plosion.Ilssontne´anmoinsaccessiblesa` toutlecteurposse´dantunebonnefamilia- rite´ avec les fonctions analytiques d’une variable complexe. Les divers outils sont soigneusement mis en place, progressivement et avec beaucoup d’exemples. C’est unebellere´ussite. A` Toulouse,le16mai2014, Jean-PierreRamis Preface to the Three Volumes Thisthree-volumeseriesaroseoutoflecturenotesforthecourseswegavetogether ataCIMPA1 schoolinLima,Peru,inJuly2008.Sincethen,thesenoteshavebeen used and developed in graduate courses held at our respective institutions, that is, the universities of Angers, Nantes, Strasbourg (France) and the Scuola Normale SuperiorediPisa(Italy).Theoriginalnoteshavenowgrownintoself-containedin- troductionstoproblemsraisedbyanalyticcontinuationandthedivergenceofpower seriesinonecomplexvariable,especiallywhenrelatedtodifferentialequations. A classical way of solving an analytic differential equation is the power series method,whichsubstitutesapowerseriesfortheunknownfunctionintheequation, then identifies the coefficients. Such a series, if convergent, provides an analytic solution to the equation. This is what happens at an ordinary point, that is, when we have an initial value problem to which the Cauchy-Lipschitz theorem applies. Otherwise, at a singular point, even when the method can be applied the resulting series most often diverges; its connection with “actual” local analytic solutions is notobviousdespiteitsdeeplinktotheequation. The hidden meaning of divergent formal solutions was already pondered in the nineteenthcentury,afterCauchyhadclarifiedthenotionsofconvergenceanddiver- genceofseries.Forordinarylinear differentialequations,ithasbeenknownsince thebeginningofthetwentiethcenturyhowtodetermineafullsetoflinearlyinde- pendentformalsolutions2atasingularpointintermsofafinitenumberofcomplex powers, logarithms, exponentials and power series, either convergent or divergent. Theseformalsolutionscompletelydeterminethelineardifferentialequation;hence, they contain all information about the equation itself, especially about its analytic solutions.Extractingthisinformationfromthedivergentsolutionswastheunderly- 1 Centre International de Mathe´matiques Pures et Applique´es, or ICPAM, is a non-profit inter- nationalorganizationfoundedin1978inNice,France.Itpromotesinternationalcooperationin highereducationandresearchinmathematicsandrelatedsubjectsforthebenefitofdeveloping countries.ItissupportedbyUNESCOandIMU,andmanynationalmathematicalsocietiesover theworld. 2 Onesaysaformalfundamentalsolution. ix
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