Twenty-One Lectures on Complex Analysis. A first Course

189 Pages·2017·2.784 MB·english

by  Alexander Isaev

Alexander Isaev
Twenty-One Lectures
on Complex Analysis
A First Course
Alexander Isaev
Mathematical Sciences Institute
Australian National University
Acton, Aust Capital Terr, Australia

ISSN 1615-2085 ISSN 2197-4144 (electronic)
Springer Undergraduate Mathematics Series
ISBN 978-3-319-68169-6 ISBN 978-3-319-68170-2 (eBook)
DOI 10.1007/978-3-319-68170-2

Library of Congress Control Number: 2017958438

Mathematics Subject Classification (2010): 97Ixx, 97I80

© Springer International Publishing AG 2017
Preface
ThisbookhasgrownoutofanundergraduatecoursethatIhavetaughtattheAus-
tralian National University (ANU) for over 20 years. The course is one-semester
long, which means that it runs for a total of 12 weeks. In each week I teach two
lectures, where a lecture is defined as a 100-minute class with a 5-minute break
in the middle. This lecture format is common in Europe but can be implemented
almost everywhere; for example, at the ANU—where a normal teaching period is
50minutes—Isimplyreservetwoconsecutiveperiodsforeverylecture.A12-week
semestercaninprincipleaccommodate24lecturesofthiskindbutthecoursema-
terial only occupies 21, with the remaining time spent on discussing assignments,
review,etc.
While I was transforming my lecture notes into a book, I decided to keep the
splitting of the course material into 21 lectures. This approach is unusual as most
authorswouldorganisethecontentintochapters,witheachchapteraccommodating
aparticulartopic.However,Ifindthatdividingthematerialaccordingtothewayit
ispresentedatthelectureshasseveraladvantagescomparedtothetraditionaltopic-
basedbookcomposition.Indeed,firstofall,thelecture-basedorganisationensures
that the content is partitioned into (approximately) equal pieces, so that none of
them stands out and looks intimidating to the students at least as far as the length
isconcerned.Thisissuebecomesparticularlyimportantforthosewhowishtouse
thebookforself-studyandwouldliketokeepacloseeyeontheiroverallprogress.
Secondly, the lecture-based format guarantees that the students get more training
forthemoreadvancedtopics,whicharespreadoverseverallectures.Indeed,each
lecturehasitsownuniquesetofexercises,andthestudentsarestronglyencouraged
to do at least some of them before moving on. It is then automatic that the harder
the topic, the more exercises one is expected to do to go through it. It should also
bementionedthatsomeoftheexercisesforeachlectureserveasapreparationfor
thefollowingone.Thirdly,thelecture-basedsplit-upgivesclearteachingguidelines
to the instructor, at the same time allowing for the possibility of re-arranging the
materialaccordingtotheirowntaste.
Asthebookcoversaone-semestercourse,itisshorterthanmostcomplexanaly-
sistexts(approximately200pages).Itiswell-knownthatmanystudentsareintimi-
datedbylonglargebooks,sothisshorterone—whichcontainsexactlythematerial
that needs to be learned in a one-semester course—is expected to have a broader
appeal. Another feature of the book that the students may like is a reader-friendly
conversational style of writing. For instance, there are plenty of fully worked-out
examples and textual explanations of formal statements, with plain words system-
atically used in the formulations of theorems, propositions, etc. Furthermore, the
reader is invited toparticipate in the exposition by fillingin various details of for-
malargumentsasindicatedbyparenthesisedexpressions,e.g.,(check!),(explain!),
(provide details!). The proofs of some of the statements are left as homework. In
fact, doing weekly homework is strongly encouraged with plenty of exercises to
choosefromattheendofeachlecture.Notethattheexerciseshaveavaryingdegree
of difficulty to accommodate different cohorts of students and range from routine
questionstoratherhardproblems.
Although the choice of topics covered in the book may appear to be standard,
this is more than just a book on complex analysis since it discusses concepts that
lie outside the scope of a typical complex analysis course, such as homotopy and
algebraicpropertiesofgroupsofconformaltransformations.Infact,theexposition
isnon-standardinthatthecentralresult,fromwhichmostofthematerialfollows,
is Cauchy’s Independence of Homotopy Theorem (in this regard, I was certainly
influencedbyA.VitushkinfromwhomItookmyfirstcomplexanalysiscourseand
wholaterbecamemyPhDthesisadviser).Expositionsbasedontheabovetheorem
arehardtocomeby,andthosethatIamawareofdonotsatisfyme,oftenbecause
of lack of rigour. At the same time, homotopy independence allows one to have a
nicecleanderivationofCauchy’sIntegralTheoremandCauchy’sIntegralFormula
(seeLecture10).ThisisonereasonwhyIhavedecidedtowritemyownbook.
Another instance of a non-standard approach to exposition is the proof of the
Fundamental Theorem of Algebra given in Lecture 1. The fact that this important
resultcanbeobtainedsoearlyinthebookandinsuchanelementarywaydemon-
strates the power of complex numbers and sets the tone for the entire course. The
book concludes with a proof of another major milestone, the Riemann Mapping
Theorem,whichisrarelypartofaone-semesterundergraduatecourse.Istressthat
the exposition is almost entirely self-contained, with only a handful of statements
includedwithoutproof.
Certainly, the nice extras incorporated in the book as described above come at
a price: one has to possess a certain degree of mathematical maturity in order to
understand and appreciate them. Namely, the students are required to have done a
prerequisitecourseinrealanalysisandmetricspaces,towhichIreferformostfacts
mentioned without proof. If one does not assume such a course (as is the case at
some universities in the US), the instructor may decide to exclude certain topics,
e.g.,theproofoftheRiemannMappingTheorem,andusethebookforteachinga
slightlylower-levelcourse.Theinstructorwillfindthatfittingevensuchareduced
amountofmaterialinonesemesterrequiresasignificanteffort,sooneshouldnot
be afraid of running out of content. Alternatively, although this book is primarily
aimedatundergraduates,itcanbeusedtoteachagraduatecoursetostudentswho
havetherightprerequisites.
Beforeproceeding,Iwouldliketosayanotherwordabouttheexercises.Many
of them are of my own making, but over the years I have also collected a good
number of interesting unpublished problems informally communicated to me by
various people, most of all by A. Vitushkin and E. Chirka. Some of the problems
that I learned from them are included in the book, and I am grateful to these two
mathematiciansfortheirgenerouscontributions.
Canberra,
June2017 AlexanderIsaev
Contents
1 ComplexNumbers.TheFundamentalTheoremofAlgebra ......... 1
Exercises ...................................................... 7
2 R-andC-Differentiability....................................... 9
Exercises ...................................................... 14
3 The Stereographic Projection. Conformal Maps. The Open
MappingTheorem ............................................. 17
Exercises ...................................................... 22
4 ConformalMaps(Continued).Mo¨biusTransformations............ 25
Exercises ...................................................... 30
5 Mo¨biusTransformations(Continued).GeneralisedCircles.Symmetry 33
Exercises ...................................................... 38
6 DomainsBoundedbyPairsofGeneralisedCircles.Integration...... 41
Exercises ...................................................... 46
7 Primitives Along Paths. Holomorphic Primitives on a Disk.
Goursat’sLemma .............................................. 49
Exercises ...................................................... 54
8 ProofofLemma7.2.Homotopy.TheRiemannMappingTheorem ... 57
Exercises ...................................................... 64
9 Cauchy’sIndependenceofHomotopyTheorem.JordanDomains.... 67
Exercises ...................................................... 74
10 Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s
IntegralFormula............................................... 77
Exercises ...................................................... 84
11 Morera’sTheorem.PowerSeries.Abel’sTheorem.DiskandRadius
ofConvergence ................................................ 87
Exercises ...................................................... 93
12 PowerSeries(Cont’d).ExpansionofaHolomorphicFunction.The
UniquenessTheorem ........................................... 97
Exercises ......................................................103
13 Liouville’sTheorem.LaurentSeries.IsolatedSingularities..........107
Exercises ......................................................114
14 Isolated Singularities (Continued). Poles and Zeroes. Isolated
Singularitiesat¥...............................................117
Exercises ......................................................124
15 IsolatedSingularitiesat¥(Continued).Residues.Cauchy’sResidue
Theorem ......................................................127
Exercises ......................................................133
16 Residues(Continued).ContourIntegration.TheArgumentPrinciple 137
Exercises ......................................................143
17 The Argument Principle (Cont’d). Rouche´’s Theorem. The
MaximumModulusPrinciple....................................147
Exercises ......................................................152
18 Schwarz’s Lemma. (Pre)-Compactness. Montel’s Theorem.
Hurwitz’sTheorem.............................................157
Exercises ......................................................163
19 AnalyticContinuation ..........................................167
Exercises ......................................................172
20 AnalyticContinuation(Continued).TheMonodromyTheorem......175
Exercises ......................................................181
21 Proof of Theorem 8.3. Conformal Transformations of Simply-
ConnectedDomains ............................................183
Exercises ......................................................189
Index .............................................................191
Lecture 1
Complex Numbers. The Fundamental Theorem
of Algebra
Thefieldofcomplexnumbers,orthecomplexplane,denotedbyC,isjusttheusual
EuclideanplaneR2endowedwiththeadditionaloperationofmultiplicationofvec-
torsdefinedasfollows:for(x ;y )and(x ;y )inR2let
1 1 2 2
(x ;y )(cid:1)(x ;y ):=(x x (cid:0)y y ;x y +x y ):
1 1 2 2 1 2 1 2 1 2 2 1
Noticethatify =0,theaboveoperationissimplythescalingofthevector(x ;y )
1 2 2
byx .
1
Consider the standard basis e :=(1;0) and e :=(0;1) in R2 and write it as
1 2
1:=e ,i:=e .Thenwehave
1 2
(x;y)=xe +ye =x1+yi;
1 2
andweabbreviatethelatterexpressionasx+iy.Clearly,
1(cid:1)(x+iy)=x+iy 8x;y2R;
i2=i(cid:1)i=(cid:0)1:
Usingtheabovetworules,itiseasytoremembertheformulaformultiplicationby
formallymultiplyingtheexpressions(x +iy )and(x +iy ):
1 1 2 2
(x ;y )(cid:1)(x ;y )=(x +iy )(cid:1)(x +iy )=x x +ix y +iy x +i2y y =
1 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2
(x x (cid:0)y y )+i(x y +x y ):
1 2 1 2 1 2 2 1
Itisnothardtoseethattheabovemultiplicationoperation,calledcomplexmulti-
plication,satisfiestheusualfieldaxioms.Inparticular,aswewillexplainbelow,for
everynon-zerocomplexnumberthereexistsareciprocalnumber.Inwhatfollows,
we omit the symbol (cid:1) and abbreviate (x +iy )(cid:1)(x +iy ) as (x +iy )(x +iy ).
1 1 2 2 1 1 2 2
Whenwemultiplyacomplexnumber(x+iy)byitself,weusethesymbol(x+iy)2.
Similarly,multiplicationofncopiesof(x+iy)isexpressedbythesymbol(x+iy)n.
Weusuallywriteacomplexnumberasz=x+iy,wherexiscalledtherealpartof
z (denotedbyRez)andytheimaginarypartofz (denotedbyImz).IfImz=0,the
© Springer International Publishing AG 2017 1
A. Isaev, Twenty-One Lectures on Complex Analysis, Springer Undergraduate
Mathematics Series, DOI 10.1007/978-3-319-68170-2_1
2 1 ComplexNumbers.TheFundamentalTheoremofAlgebra
complexnumberziscalledreal.Wethinkofthesetofallsuchnumbers(namely,the
x-axis)asacopyofRlyinginC.Clearly,forrealnumberscomplexmultiplication
coincides with the usual (real) multiplication, so C can be viewed as an extension
ofR.
If Rez=0, then z is said to be imaginary. The set of all imaginary numbers
(namely,they-axis)isgeometricallyanothercopyofRlyinginC.Note,however,
thatthiscopyisnotclosedwithrespecttocomplexmultiplicationastheproductof
twoimaginarynumbersisreal.
We call x(cid:0)iy the complex number conjugate to z=x+iy and denote it by z¯.
OnehasRez¯=Rez,Imz¯=(cid:0)Imzand
z+z¯ z(cid:0)z¯
Rez= ; Imz=(cid:0)i :
2 2
Thenumberz¯shouldnotbeconfusedwiththecomplexnumberoppositetoz,which
is(cid:0)z=(cid:0)x(cid:0)iy.
p
Define the modulus, or absolute value, of z by jzj:= zz¯. As zz¯=x2+y2, it
followsthatjzjistheusualEuclideanlengthofthevector(x;y)inR2.Themodulus
hasthefamiliarpropertiesofthemodulusofrealnumbers,e.g.,
(cid:12) (cid:12)
jz +z j(cid:20)jz j+jz j; jz +z j(cid:21)(cid:12)jz j(cid:0)jz j(cid:12)
1 2 1 2 1 2 (cid:12) 1 2 (cid:12)
(check!). The distance between any two complex numbers z , z is just the Eu-
1 2
clideandistancejz (cid:0)z j,whichturnsCintoametricspace.Below,convergenceof
1 2
sequences of complex numbers, limits of C-valued functions, continuity, etc., will
bemostlyunderstoodwithrespecttothismetricspacestructure.
Next,foranon-zeroz,thereciprocalof z,ifexists,istheuniquecomplexnumber
1
w such that wz=1. The reciprocal of z is denoted by either z(cid:0)1 or . Using the
z
conjugate number and the modulus, it is easy to see that z(cid:0)1 exists for every non-
zeroz;infact,onecancomputez(cid:0)1byscalingthevectorz¯asfollows:
1
z(cid:0)1= z¯:
jzj2
Wecannowintroducetheratiooftwocomplexnumbersz ,z ,withz 6=0,as
1 2 2
z
1 :=z z(cid:0)1:
z 1 2
2
Further,intermsofthemodulusfunction,define
D(a;r):=fz2C:jz(cid:0)aj<rg
to be the open disk of radius 0<r<¥ centred at a. We write D for D(0;r) and
r
denotetheunitdisk D simplybyD.Also,foropendisksofzeroandinfiniteradii
1
wesetD(a;0):=0/ andD(a;¥):=C.
Next,onecanexpressanyz6=0as
1 ComplexNumbers.TheFundamentalTheoremofAlgebra 3
z=jzj(cosj;sinj)=jzj(cosj+isinj);
where0(cid:20)j<2p istheanglebetweenthehalf-line R :=fx2R:x(cid:21)0gandthe
+
half-lineemanatingfrom0andpassingthroughz,calculatedintheanti-clockwise
directionstartingwithR .Thisnumberiscalledtheargumentofz andisdenoted
+
byargz(seeFig.1.1).Then
z=jzj(cosargz+isinargz)=jzjeiargz;
whereforanyrealnumbert weseteit :=cost+isint.
y
z
arg z
0 x
Fig.1.1
Ingeneral,foranyz=x+iy2Cdefine
ez:=ex(cosy+isiny)=exeiy:
Thefunctioneziscalledtheexponentialfunction.Ithasthefollowingfamiliarprop-
erty:
Proposition1.1.Forz1;z22Cwehaveez1+z2 =ez1ez2.
Proof. Bythedefinitionoftheexponentialfunctionwewrite
ez1+z2 =ex1+x2(cos(y +y )+isin(y +y )):
1 2 1 2
Ontheotherhand,
ez1ez2 =ex1(cosy1+isiny1)ex2(cosy2+isiny2)=
ex1+x2((cosy1cosy2(cid:0)siny1siny2)+i(cosy1siny2+siny1cosy2))=
ex1+x2(cos(y1+y2)+isin(y1+y2));
whereweutilisedtheusualtrigonometricidentitiesforcosandsinofsumsofargu-
ments. tu