Complex Analysis, Riemann Surfaces and Integrable Systems PDF
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Moscow Lectures 3 Sergey M. Natanzon Complex Analysis, Riemann Surfaces and Integrable Systems Moscow Lectures Volume 3 SeriesEditors LevD.Beklemishev,Moscow,Russia VladimirI.Bogachev,Moscow,Russia BorisFeigin,Moscow,Russia ValeryGritsenko,Moscow,Russia YulyS.Ilyashenko,Moscow,Russia DmitryB.Kaledin,Moscow,Russia AskoldKhovanskii,Moscow,Russia IgorM.Krichever,Moscow,Russia AndreiD.Mironov,Moscow,Russia VictorA.Vassiliev,Moscow,Russia Managingeditor AlexeyL.Gorodentsev,Moscow,Russia Moreinformationaboutthisseriesathttp://www.springer.com/series/15875 Sergey M. Natanzon Complex Analysis, Riemann Surfaces and Integrable Systems 123 SergeyM.Natanzon HSEUniversity Moscow,Russia TranslatedfromtheRussianbyNataliaTsilevich.:OriginallypublishedasКомплексный анализ, римановы поверхности и интегрируемыесистемыbyMCCME,2018. ISSN2522-0314 ISSN2522-0322 (electronic) MoscowLectures ISBN978-3-030-34639-3 ISBN978-3-030-34640-9 (eBook) https://doi.org/10.1007/978-3-030-34640-9 MathematicsSubjectClassification(2010):30C35,30F10,32G15,37K10,37K20 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Cover illustration: https://www.istockphoto.com/de/foto/panorama-der-stadt-moskau-gm490080014- 75024685,withkindpermission ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface to the Book Series Moscow Lectures YouholdavolumeinatextbookseriesofSpringerNaturededicatedtotheMoscow mathematical tradition. Moscow mathematics has very strong and distinctive fea- tures.Thereareseveralreasonsforthis,allofwhichgobacktogoodandbadaspects of Soviet organization of science. In the twentieth century, there was a veritable galaxyofgreatmathematiciansinRussia,whileitsohappenedthattherewereonly few mathematicalcenters in which these experts clustered. A major one of these, andperhapsthemostinfluential,wasMoscow. There are three major reasons for the spectacular success of Soviet mathematics: 1. Significant support from the governmentand the high prestige of science as a profession.Bothfactorswererelatedtotheprocessofrapidindustrializationin theUSSR. 2. Doing research in mathematics or physics was one of very few intellectual activities thathad nomandatoryideologicalcontent.Many would-becomputer scientists,historians,philosophers,oreconomists(andevenartistsormusicians) becamemathematiciansorphysicists. 3. TheIronCurtainpreventedinternationalmobility. These are specific factors that shaped the structure of Soviet science. Certainly, factors(2)and(3)aremoreonthenegativesideandcannotreallybecalledfavorable but they essentially came together in combination with the totalitarian system. Nowadays, it would be impossible to find a scientist who would want all of the threefactorstobebackintheirtotality.Ontheotherhand,thesefactorsleftsome positiveandlonglastingresults. An unprecedented concentration of many bright scientists in few places led eventuallytothedevelopmentofaunique“Sovietschool”.Ofcourse,mathematical schools in a similar sense were formed in other countries too. An example is the Frenchmathematicalschool,whichhasconsistentlyproducedfirst-rateresultsover alongperiodoftimeandwhereanextensivedegreeofcollaborationtakesplace.On the other hand, the British mathematicalcommunitygave rise to many prominent successes but failed to form a “school” due to a lack of collaborations. Indeed, a v vi PrefacetotheBookSeriesMoscowLectures school as such is not only a large group of closely collaborating individuals but alsoagroupknittightlytogetherthroughstudent-advisorrelationships.IntheUSA, whichiscurrentlytheworldleaderintermsofthelevelandvolumeofmathematical research, the level of mobility is very high, and for this reason there are no US mathematical schools in the Soviet or French sense of the term. One can talk not only about the Soviet school of mathematics but also, more specifically, of the Moscow, Leningrad, Kiev, Novosibirsk, Kharkov, and other schools. In all these places, there were constellationsof distinguished scientists with large numbersof students,conductingregularseminars.Thesedistinguishedscientistswereoftennot merelyadvisorsandleaders,butoftentheyeffectivelybecamespiritualleadersina verygeneralsense. AcharacteristicfeatureoftheMoscowmathematicalschoolisthatitstressesthe necessity for mathematicians to learn mathematics as broadly as they can, rather thanfocusingonanarrowfieldinordertogetimportantresultsassoonaspossible. The Moscow mathematical school is particularly strong in the areas of alge- bra/algebraicgeometry,analysis,geometryandtopology,probability,mathematical physics and dynamical systems. The scenarios in which these areas were able to developinMoscowhavepassedintohistory.However,itispossibletomaintainand develop the Moscow mathematical tradition in new formats, taking into account modern realities such as globalization and mobility of science. There are three recently created centers—the Independent University of Moscow, the Faculty of Mathematics at the National Research University Higher School of Economics (HSE) and the Center for Advanced Studies at Skolkovo Institute of Science and Technology(SkolTech)—whosemissionistostrengthentheMoscowmathematical traditioninnewways.HSEandSkolTechareuniversitiesofferingofficiallylicensed fulltime educationalprograms.Mathematicalcurriculaat these universitiesfollow not only the Russian and Moscow tradition but also new global developments in mathematics. Mathematical programs at the HSE are influenced by those of the Independent University of Moscow (IUM). The IUM is not a formal university; it is rather a place where mathematicsstudentsof differentuniversitiescan attend specialtopicscoursesaswellascourseselaboratingthecorecurriculum.TheIUM was the main initiator of the HSE Faculty of Mathematics. Nowadays, there is a closecollaborationbetweenthetwoinstitutions. While attempting to further elevate traditionally strong aspects of Moscow mathematics, we do not reproduce the former conditions. Instead of isolation and academic inbreeding, we foster global sharing of ideas and international cooperation. An important part of our mission is to make the Moscow tradition ofmathematicsatauniversitylevelapartofglobalcultureandknowledge. The“MoscowLectures”seriesservesthisgoal.Ourauthorsaremathematicians of different generations. All follow the Moscow mathematical tradition, and all teach or have taught university courses in Moscow. The authors may have taught mathematics at HSE, SkolTech, IUM, the Science and Education Center of the Steklov Institute, as well as traditionalschools like MechMath in MGU or MIPT. Teaching and writing styles may be very different. However, all lecture notes are PrefacetotheBookSeriesMoscowLectures vii supposedtoconveyalivedialogbetweentheinstructorandthestudents.Notonly personalitiesofthelecturersareimprintedinthesenotes,butalsothoseofstudents. We hope that expositions published within the “Moscow lectures” series will provideclearunderstandingofmathematicalsubjects,usefulintuition,andafeeling oflifeintheMoscowmathematicalschool. Moscow,Russia IgorM.Krichever VladlenA.Timorin MichaelA.Tsfasman VictorA.Vassiliev Introduction This book is based on the interrelated courses in complex analysis, the theory of Riemann surfaces, and the theory of integrable systems repeatedly taught by the author at the Independent University of Moscow and at the Department of MathematicsoftheHigherSchoolofEconomics.Theonlyprerequisiteforreading itisabasicknowledgeofcalculuscoveredinthefirst2yearsofundergraduatestudy (see, e.g., [19]). The theoreticalmaterialis complementedby exercisesof various degreesofdifficulty. The first two chapters are devoted to the classical theory of holomorphic and meromorphic functions and mostly correspond to a standard course in complex analysis. The importance of this theory lies in the fact that the language of holomorphic and meromorphic functions is used to state fundamental laws of physics. The condition for a function to be holomorphic (i.e., to have a complex derivative) turns out to be much more restrictive than the condition to have a real derivative. Holomorphic functions have a number of nice general properties, the most important of which is that the global properties of a function are to a largeextentdeterminedbyits localproperties.Thisallowsoneto makeimportant predictionsaboutscientificphenomenarelyingonlocalpropertiesofaprocess. Then we prove the classical Riemann theorem, which says that an arbitrary proper simply connected domain in the complex plane can be mapped onto the standard unit disk by a one-to-one conformal map. Such maps are described by biholomorphicfunctions.Here,wegivetheclassicalproofoftheRiemanntheorem. Unfortunately,itgivesnorecipeforconstructingthedesiredmap.However,themap itself, which sends an arbitrary domain to the disk, plays a key role in important applicationsofmathematics(hydromechanics,aerodynamics,andevenoilandgas industry[28]).Significantprogressinthecomputationofthedesiredmapwasmade inthiscenturyusingthetheoryofharmonicfunctionsandintegrablesystems.This newtheoryisconsideredinthelastchapterofthebook. Chapter 4 is devoted to the theory of harmonic functions. It is closely related to the theory of holomorphicfunctions and is extensively used in various applied problems. The central object of study here is the Green’s function of an arbitrary domainanditsapplicationtothesolutionoftheDirichletproblem. ix
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