shabat b.v. introduction to complex analysis (excerpts translated by l.ryzhik, 2003)(111s)_mcc_

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1、IntroductiontoComplexAnalysis-excerptsB.V.ShabatJune2,20032Chapter1TheHolomorphicFunctionsWebeginwiththedescriptionofcomplexnumbersandtheirbasicalgebraicproperties.Wewillassumethatthereaderhadsomepreviousencounterswiththecomplexnumbersandwillbefairlybrief,withtheemphasisonsomes

2、pecificsthatwewillneedlater.1TheComplexPlane1.1ThecomplexnumbersWeconsiderthesetCofpairsofrealnumbers(x,y),orequivalentlyofpointsontheplaneR2.Twovectorsz=(x,x)andz=(x,y)areequalifandonlyifx=x11222212andy1=y2.Twovectorsz=(x,y)and¯z=(x,−y)thataresymmetrictoeachotherwithrespecttoth

3、ex-axisaresaidtobecomplexconjugatetoeachother.Weidentifythevector(x,0)witharealnumberx.WedenotebyRthesetofallrealnumbers(thex-axis).Exercise1.1Showthatz=¯zifandonlyifzisarealnumber.WeintroducenowtheoperationsofadditionandmultiplicationonCthatturnitintoafield.Thesumoftwocomplexnu

4、mbersandmultiplicationbyarealnumberλ∈RaredefinedinthesamewayasinR2:(x1,y1)+(x2,y2)=(x1+x2,y1+y2),λ(x,y)=(λx,λy).Thenwemaywriteeachcomplexnumberz=(x,y)asz=x·1+y·i=x+iy,(1.1)wherewedenotedthetwounitvectorsinthedirectionsofthexandy-axesby1=(1,0)andi=(0,1).Youhavepreviouslyencounter

5、edtwowaysofdefiningaproductoftwovectors:theinnerproduct(z1·z2)=x1x2+y1y2andtheskewproduct[z1,z2]=x1y2−x2y1.However,noneofthemturnCintoafield,and,actuallyCisnotevenclosedunderthese34CHAPTER1.THEHOLOMORPHICFUNCTIONSoperations:boththeinnerproductandtheskewproductoftwovectorsisanumbe

6、r,notavector.ThisleadsustointroduceyetanotherproductonC.Namely,wepostulatethati·i=i2=−1anddefinezzasavectorobtainedbymultiplicationofx+iyand1211x+iyusingtheusualrulesofalgebrawiththeadditionalconventionthati2=−1.22Thatis,wedefinez1z2=x1x2−y1y2+i(x1y2+x2y1).(1.2)Moreformallywemayw

7、rite(x1,y1)(x2,y2)=(x1x2−y1y2,x1y2+x2y1)butwewillnotusethissomewhatcumbersomenotation.Exercise1.2Showthattheproductoftwocomplexnumbersmaybewrittenintermsoftheinnerproductandtheskewproductasz1z2=(¯z1·z2)+i[¯z1,z2],wherez¯1=x1−iy1isthecomplexconjugateofz1.Exercise1.3Checkthatthep

8、roduct(1.2)turnsCintoafield,thatis,thedistributive,comm