complex analysis.pdf

COMPLEXANALYSIS 1

DouglasN.Arnold 2

References:

JohnB.Conway,FunctionsofOneComplexVariable,Springer-Verlag,1978.

LarsV.Ahlfors,ComplexAnalysis,McGraw-Hill,1966.

RaghavanNarasimhan,ComplexAnalysisinOneVariable,Birkh¨auser,1985.

CONTENTS

I.TheComplexNumberSystem.............................................2

II.ElementaryPropertiesandExamplesofAnalyticFns................3

Dierentiabilityandanalyticity...........................................4

TheLogarithm............................................................6

Conformality..............................................................6

Cauchy–RiemannEquations...............................................7

M¨obiustransformations...................................................9

III.ComplexIntegrationandApplicationstoAnalyticFns.............11

Localresultsandconsequences...........................................12

HomotopyofpathsandCauchy’sTheorem...............................14

WindingnumbersandCauchy’sIntegralFormula.........................15

Zerocounting;OpenMappingTheorem..................................17

Morera’sTheoremandGoursat’sTheorem...............................18

IV.SingularitiesofAnalyticFunctions....................................19

Laurentseries............................................................20

Residueintegrals.........................................................23

V.Furtherresultsonanalyticfunctions.................................26

ThetheoremsofWeierstrass,Hurwitz,andMontel.......................26

Schwarz’sLemma........................................................28

TheRiemannMappingTheorem.........................................29

ComplementsonConformalMapping....................................31

VI.HarmonicFunctions......................................................32

ThePoissonkernel.......................................................33

SubharmonicfunctionsandthesolutionoftheDirichletProblem.........36

TheSchwarzReﬂectionPrinciple.........................................39

1 Theselecturenoteswerepreparedfortheinstructor’spersonaluseinteachingahalf-semestercourse

oncomplexanalysisatthebeginninggraduatelevelatPennState,inSpring1997.Theyarecertainlynot

meanttoreplaceagoodtextonthesubject,suchasthoselistedonthispage.

2 DepartmentofMathematics,PennStateUniversity,UniversityPark,PA16802.

Web:http://www.math.psu.edu/dna/.

I.TheComplexNumberSystem

Risaﬁeld.Forn>1,R n isavectorspaceoverR,soisanadditivegroup,butdoesn’t

haveamultiplicationonit.WecanendowR 2 withamultiplicationby

(a,b)(c,d)=(ac−bd,bc+ad).

UnderthisdeﬁnitionR 2 becomesaﬁeld,denotedC.Notethat(a/(a 2 +b 2 ),−b/(a 2 +b 2 ))

isthemultiplicativeinverseof(a,b).(Remark:itisnotpossibletoendowR n withaﬁeld

structureforn>2.)Wedenote(0,1)byiandidentifyx2Rwith(x,0),soRC.Thus

(a,b)=a+bi,a,b2R.Notethati 2 =−1.CisgeneratedbyadjoiningitoRandclosing

underadditionandmultiplication.Itisremarkablethattheadditionofiletsusnotonly

solvetheequationx 2 +1=0,buteverypolynomialequation.

Foraandbrealandz=a+biwedeﬁneRez=a,Imz=b,¯z=a−bi,and

|z|=(a 2 +b 2 ) 1/2 .Then

Rez=(z+¯z)/2, Imz=(z−¯z)/(2i),

|z| 2 =z¯z, 1

z/w=¯z/¯w, |z+w||z|+|w|.

Themap7!(cos,sin)deﬁnesa2-periodicmapofthereallineontotheunit

circleinR 2 .Incomplexnotationthismapis7!cis:=cos+isin.Everynonzero

complexnumbercanbewrittenasrciswherer>0isuniquelydeterminedand2R

isuniquelydeterminedmodulo2.Thenumber0isequaltorciswherer=0and

isarbitrary.Therelationz=rcisdeterminestherelationsz7!rwhichissimplythe

functionr=|z|andz7!.Thelatterisdenoted=arg.Notethatforz6=0,argis

determinedmodulo2(whilearg0isarbitrary).Wecannormalizeargbyinsistingthat

argz2(−,].Notethatifz 1 =rcis 1 andz 2 =rcis 2 thenz 1 z 2 =r 1 r 2 cis( 1 + 2 ).

Thelatterformulajustencapsulatestheformulaforthesineandcosineofasum,andgives

argz 1 z 2 =argz 1 +argz 2 .Inparticular,ircis=rcis(+/2),somultiplicationbyiis

justtheoperationofrotationby/2inthecomplexplane.Multiplicationbyanarbitrary

complexnumberrcisisjustrotationbyargfollowedby(orprecededby)dilationbya

factorr.Further,z n =r n cis(n).Everynonzeroz2Cadmitsndistinctnthroots:the

z = ¯z

|z| 2 ,

z±w=¯z±¯w, zw=¯z¯w,

nthrootsofrcisare n p rcis[(+2k)/n],k=0,1,...,n.

Linesandcirclesintheplane.Circlesgivenby|z−a|=rwherea2Cisthecenter

andr>0istheradius.If06=b2Cthenthelinethroughtheorigininthedirection

bisthesetofallpointsoftheformtb,t2R,orallzwithIm(z/b)=0.Ift2R

andc>0then(t+ci)b=tb+cibrepresentsapointinthehalfplanetotheleftof

bdeterminedbythelinetb,i.e.,{z:Im(z/b)>0}istheequationofthathalf-plane.

Similarly,{z:Im[(z−a)/b]>0}isthetranslationofthathalf-planebya,i.e.,thehalf-

planedeterminedbythelinethroughaparalleltobandinthedirectiontotheleftof

Stereographicprojectiondeterminesaone-to-onecorrespondencebetweentheunit

sphereinR 3 minusthenorth-pole,S,andthecomplexplaneviathecorrespondence

z$ x 1 +ix 2

1−x 3 ,

1+|z| 2 , x 2 = 2Imz

1+|z| 2 , x 3 = |z| 2 −1

|z|+1 .

IfwedeﬁneC 1 =C[{1},thenwehaveaone-to-onecorrespondencebetweenSand

C 1 .ThisallowsustodeﬁneametriconC 1 ,whichisgivenby

d(z 1 ,z 2 )= 2|z 1 −z 2 |

p 1+|z| 2 .

II.ElementaryPropertiesandExamplesofAnalyticFunctions

Forz6=1, P N n=0 z n =(1−z N+1 )/(1−z).Thereforethegeometricseries P 1 n=0 z n

converges(to1/(1−z))if|z|<1.Itclearlydiverges,infactitstermsbecomeunbounded,

if|z|>1.

WeierstrassM-Test.LetM 0 ,M 1 ,...bepositivenumberswith P M n <1andsuppose

thatf n :X!CarefunctionsonsomesetXsatisfyingsup x2X |f n (x)|M n .Then

R =limsup|a n | 1/n .

Then(1)foranya2Cthepowerseries P 1 n=0 a n (z−a) n convergesabsolutelyforall

|z−a|<Randitconvergesabsolutelyanduniformlyonthedisk|z−a|rforallr<R.

(2)Thesequencea n (z−a) n isunboundedforall|z−a|>R(andhencetheseriesis

certainlydivergent).

Thusweseethatthesetofpointswhereapowerseriesconvergesconsistsofadisk

|z−a|<Randpossiblyasubsetofitsboundary.Riscalledtheradiusofconvergenceof

itsseries.ThecaseR=1isallowed.

Proofoftheorem.Foranyr<RweshowabsoluteuniformconvergenceonD r ={|z−a|

r}.Choose˜r2(r,R).Then,1/˜r>limsup|a n | 1/n ,so|a n | 1/n <1/˜rforallnsuciently

large.Forsuchn,|a n |<1/˜r n andso

sup

z2D r

|a n (z−a) n |<(r/˜r) n .

Since P (r/˜r) n <1wegettheabsoluteuniformconvergenceonD r .

If|z−a|=r>R,take˜r2(R,r).Thenthereexistnarbitrarilylargesuchthat

|a n | 1/n 1/˜r.Then,|a n (z−a) n |(r/˜r) n ,whichcanbearbitrarilylarge.

x 1 = 2Rez

p (1+|z 1 | 2 )(1+|z 2 | 2 ) , d(z,1)= 2

P 1 n=0 f n (x)isabsolutelyanduniformlyconvergent.

Theorem.Leta 0 ,a 1 ,···2CbegivenanddeﬁnethenumberRby

thislimitistheradiusofconvergenceRof P a n (z−a) n .

Thustheseries P a n z n hastermsofincreasingmagnitude,andsocannotbeconvergent.

Thus|z|R.Thisshowsthatlim|a n /a n+1 |R.

Similarly,supposethatz<lim|a n /a n+1 |.Thenforallnsucientlylarge|a n |>|a n+1 z|

and|a n z n |>|a n+1 z n+1 |.Thustheserieshastermsofdecreasingmagnitude,andso,by

theprevioustheorem,|z|R.Thisshowsthatlim|a n /a n+1 |R.

Remark.Onthecircleofconvergence,manydierentbehaviorsarepossible. P z n diverges

forall|z|=1. P z n /ndivergesforz=1,elseconverges,butnotabsolutely(thisfollows

fromthefactthatthepartialsumsof P z n areboundedforz6=1and1/n#0). P z n /n 2

convergesabsolutelyon|z|1.Sierpinskigavea(complicated)exampleofafunction

whichdivergesateverypointoftheunitcircleexceptz=1.

Asanapplication,weseethattheseries

1 X

z n

n=0

convergesabsolutelyforallz2Candthattheconvergenceisuniformonallboundedsets.

Thesumis,bydeﬁnition,expz.

Nowsupposethat P 1 n=0 a n (z−a) n hasradiusofconvergenceR,andconsideritsformal

hasthesameradiusofconvergenceas P 1 n=0 a n (z−a) n since

(z−a)

N X

a n+1 (z−a) n =

N+1 X

a n (z−a)−a 0 ,

n=0

andsothepartialsumsontheleftandrighteitherbothdivergeforagivenzorboth

converge.Thisshows(inaroundaboutway)thatlimsup|a n+1 | 1/n =limsup|a n | 1/n =

1/R.Nowlim(n+1) 1/n =1asiseasilyseenbytakinglogs.Moreover,itiseasytoseethatif

limsupb n =bandlimc n =c>0,thenlimsupb n c n =bc.Thuslimsup|(n+1)a n+1 | 1/n =

1/R.Thisshowsthattheformalderivativeofapowerserieshasthesameradiusof

convergenceastheoriginalpowerseries.

Dierentiabilityandanalyticity.Deﬁnitionofdierentiabilityatapoint(assumes

functionisdeﬁnedinaneighborhoodofthepoint).

Mostoftheconsequencesofdierentiabilityarequitedierentintherealandcomplex

case,butthesimplestalgebraicrulesarethesame,withthesameproofs.Firstofall,

dierentiabilityatapointimpliescontinuitythere.Iffandgarebothdierentiableata

pointa,thensoaref±g,f·g,and,ifg(a)6=0,f/g,andtheusualsum,product,and

quotientruleshold.Iffisdierentiableataandgisdierentiableatf(a),thengf

isdierentiableataandthechainruleholds.Supposethatfiscontinuousata,gis

continousatf(a),andg(f(z))=zforallzinaneighborhoodofa.Thenifg 0 (f(a))exists

andisnon-zero,thenf 0 (a)existsandequals1/g 0 (f(a)).

Theorem.Ifa 0 ,a 1 ,...2Candlim|a n /a n+1 |existsasaﬁnitenumberorinﬁnity,then

Proof.Withoutlossofgeneralitywecansupposethata=0.Supposethat|z|>

lim|a n /a n+1 |.Thenforallnsucientlylarge|a n |<|a n+1 z|and|a n z n |<|a n+1 z n+1 |.

derivative P 1 n=1 na n (z−a) n−1 = P 1 n=0 (n+1)a n+1 (z−a) n .Nowclearly P n=0 a n+1 (z−a) n

Deﬁnition.Letfbeacomplex-valuedfunctiondeﬁnedonanopensetGinC.Thenf

issaidtobeanalyticonGiff 0 existsandiscontinuousateverypointofG.

Remark.WeshallprovelaterthatiffisdierentiableateverypointofanopensetinC

itisautomaticallyanalytic;infact,itisautomaticallyinﬁnitelydierentiable.Thisisof

coursevastlydierentfromtherealcase.

IfQisanarbitrarynon-emptysubsetofCwesayfisanalyticonQifitisdeﬁnedand

analyticonanopensetcontainingQ.

Wenowshowthatapowerseriesisdierentiableateverypointinitsdiskofconvergence

andthatitsderivativeisgivenbytheformalderivativeobtainedbydierentiatingterm-

by-term.Sinceweknowthatthatpowerserieshasthesameradiusofconvergence,it

followsthatapowerseriesisanalyticandinﬁnitelydierentiableinitsconvergencedisk.

Forsimplicity,andwithoutlossofgeneralityweconsiderapowerseriescenteredatzero:

issatisﬁedforallzsucientlyclosetoz 0 ,whereg(z)= P 1 n=1 na n z n−1 .Lets N (z)=

z−z 0 −g(z 0 )

P N n=0 a n z n ,R N (z)= P 1 n=N+1 a n z n .Then

z−z 0 −g(z 0 )

z−z 0 −s 0 N (z 0 )

=:T 1 +T 2 +T 3 .

+|s 0 N (z 0 )−g(z 0 )|+

R N (z)−R N (z 0 )

z−z 0

Nows 0 N (z 0 )isjustapartialsumforg(z 0 ),soforNsucientlylarge(andallz),T 2 /3.

Also,

R N (z)−R n (z 0 )

a n z n −z n 0

z−z 0 =

z−z 0 .

n=N+1

0 |a n nr n−1 .

Since P a n nr n−1 isconvergent,wehaveforNsucientlylargeandall|z|<rthen

T 3 </3.NowﬁxavalueofNwhichissucientlylargebybothcriteria.Thenthe

dierentiabilityofthepolynomials N showsthatT 1 /3forallzsucientlyclosetoz 0 .

Wethusknowthatiff(z)= P a n z n ,then,withinthediskofconvergence,f 0 (z)=

z−z 0

=|a n ||z n−1 +z n−2 z 0 +···+z n−1

P na n z n−1 ,andbyinduction,f 00 (z)= P n(n−1)a n z n−2 ,etc.Thusa 0 =f(0),a 1 =f 0 (0),

a 2 =f 00 (0)/2,a 3 =f 000 (0)/3!,etc.Thisshowsthatanyconvergentpowerseriesisthesum

ofitsTaylorseriesinthediskofconvergence:

f(z)= X f n (a)

n! (z−a) n .

Inparticular,exp 0 =exp.

f(z)= P n a n z n .SupposethattheradiusofconvergenceisRandthat|z 0 |<R.Wemust

showthatforany>0,theinequality

f(z)−f(z 0 )

s N (z)−s N (z 0 )

f(z)−f(z 0 )

Now|z 0 |<r<Rforsomer,andifwerestrictto|z|<r,wehave

a n z n −z n 0

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