Table Of ContentMoscow 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
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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.
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