复变函数英文教案

ChapterⅠComplexNumberFieldInthischapter,wesurveythealgebraicandgeometricstructureofthecomplexnumbersystem.Weassumevariouscorrespondingpropertiesofrealnumberstobeknown.Thepositiveintegernumbersystem,integernumbersystem,rationalnumbersystemandrealnumbersystemaredenotedbyandQZ,N,R,respectively.§1.1.SumsandProducts1.DefinitonofComplexNumbersAcomplexnumberisdefinedasanorderedpairofrealnumbers),(yxxandy.Itiscustomarytodenoteacomplexnumberby),(yxz,sothat),(yxz=.(1.1.1)Therealnumbersxandyarecalledtherealandimaginarypartsof,respectively;andwewritezyzxz==Im,Re.(1.1.2)Twocomplexnumbersand),(111yxz=),(222yxz=areequalwhenevertheyhavethesamerealpartsandthesameimaginaryparts.2.OperationsofComplexNumbersThesumandtheproductoftwocomplexnumbersandaredefinedasfollows:21zz+21zz),(111yxz=),(222yxz=),(),(),(21212211yyxxyxyx++=+,(1.1.3)),(),)(,(212121212211yxxyyyxxyxyx+−=.(1.1.4)3.TheRelationshipofRealNumbersandComplexNumbersNotethattheoperationsdefinedbyequations(1.1.3)and(1.1.4)becometheusualoperationsofadditionandmultiplicationwhenrestrictedtotherealnumbers:)0,()0,()0,(2121xxxx+=+,)0,()0,)(0,(2121xxxx=.Thecomplexnumbersystemis,therefore,anaturalextensionoftherealnumbersystem.4.AlternativeRepresentationofComplexNumbersFig.1-1Anycomplexnumbercanbewrittenas,anditiseasytoseethat.Hence),(yxz=),0()0,(yxz+=),0()0,)(1,0(yy=)0,)(1,0()0,(yxz+=;and,ifwethinkofarealnumberxasthecomplexnumber,thatis,weidentifyarealnumber)0,(xxwithacorrespondingcomplexnumber,andlet)0,(xidenotetheimaginarynumber(Fig.1-1))1,0(itisclearthatiyxz+=,(1.1.5)whichiscalledtherectangularformofthenumber.Thus,thecomplexnumbersystemcanbewrittenasz},:{},:),{(RRC∈+=∈=yxiyxyxyx.Also,withtheconvention,etc.,wefindthat232,zzzzzz==1)0,1()1,0)(1,0(2−=−==i.(1.1.6)Thus,theequationhasaroot012=+ziz=inC.Inviewofexpression(1.1.5),definitions(1.1.3)and(1.1.4)become)()()()(21212211yyixxiyxiyx+++=+++,(1.1.7))()())((212121212211yxxyiyyxxiyxiyx++−=++.(1.1.8)Observethattheright-handsidesoftheseequationscanbeobtainedbyformallymanipulatingthetermsontheleftreplacingby-1whenitoccurs.2iChapterComplexNumberFieldⅠ1§1.2.BasicAlgebraicPropertiesVariouspropertiesofadditionandmultiplicationofcomplexnumbersarethesameasforrealnumbers.Welistherethemorebasicofthesealgebraicpropertiesandverifysomeofthem.Mostoftheothersareverifiedintheexercises.1.Commutativelaw12211221,zzzzzzzz=+=+(1.2.1)2.Associativelaw3.Distributivelaw2121)(zzzzzzz+=+,(1.2.3)4484476Lnzzznz+++=and876Lnnzzzz=.4.IdentitiesTheadditiveidentityandthemultiplicativeidentity)0,0(0=)0,1(1=forrealnumberscarryovertotheentirecomplexnumbersystem.Thatis,zz=+0andzz=⋅1(1.2.4)foreverycomplexnumberz.Furthermore,0and1aretheonlycomplexnumberswithsuchproperties(seeExercise9).5.AdditiveinverseForeachcomplexnumber,thereisanadditiveinverse),(yxz=),(yxz−−=−,(1.2.5)6.SubtractionC∈∀−+=−212121,),(zzzzzz.(1.2.6)Soifand,then),(111yxz=),(222yxz=)()(),(2121212121yyixxyyxxzz−+−=−−=−.(1.2.7)7.MultiplicativeinverseFor0),(≠+==iyxyxz,thereisanumbersuchthat,calledthemultiplicativeinverseof.Itiseasytofindthatthemultiplicativeinverseofis1−z11=−zzziyxyxz+==),()0(,222222221≠+−++=⎟⎟⎠⎞⎜⎜⎝⎛+−+=−zyxyiyxxyxyyxxz.(1.2.8)Fromthediscussionabove,weconcludethatthesetofallcomplexnumbersbecomesafield,calledthefieldofcomplexnumbers,orthecomplexnumberfield.C§1.3.FurtherPropertiesInthissection,wementionanumberofotheralgebraicpropertiesofadditionandmultiplicationofcomplexnumbersthatfollowfromtheonesalreadydescribedinSec.1.2.Becausesuchpropertiescontinuetobeanticipated,thereadercaneasilypasstoSec.1.4withoutseriousdisruption.1.ExpunctivelawIf,theneitheror021=zz01=z02=z;orpossiblybothandequalzero.Anotherwaytostatethisresultisthatiftwocomplexnumbersandarenonzero,thensoistheirproduct.1z2z1z2z21zz2.DivisionDivisionbyanonzerocomplexnumberisdefinedasfollows:)0(212121≠=−zzzzz(1.3.1)Ifand11111),(iyxyxz+==22222),(iyxyxz+==,then)0(2222221212222212121≠+−+++=zyxyxxyiyxyyxxzz.(1.3.2)Althoughexpression(1.3.2)isnoteasytoremember,itcanbeobtainedbywriting(seeExercise4)))(())((2222221121iyxiyxiyxiyxzz−+−+=.(1.3.3)3.Usefulidentities)0(12122≠=−zzz.(1.3.4))0(122121≠⎟⎟⎠⎞⎜⎜⎝⎛=zzzzz.(1.3.5))0(1))(())((2122111121121≠==−−−−zzzzzzzzz.)0,0(11)(12121121112121≠≠⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛===−−−zzzzzzzzzz.(1.3.6))0,0(4342314321≠≠⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=zzzzzzzzzz.(1.3.7)Example.Computationssuchasthefollowingarenowjustified:.26126526265265)5)(5(55551)1)(32(111321iiiiiiiiiiiii+=+=+=+−+=++⋅−=+−=⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−4.BinomialformulaIfandareanytwocomplexnumbers,then1z2z∑=−=⎟⎟⎠⎞⎜⎜⎝⎛=+nkkknnnzzknzz02121),2,1()(K(BinomialFormula)(1.3.8)where),,2,1,0()!(!!nkknknknK=−=⎟⎟⎠⎞⎜⎜⎝⎛andwhereitisagreedthat1!0=.§1.4.ModuliItisnaturaltoassociateanynonzerocomplexnumberiyxz+=withthedirectedlinesegment,orvector,fromtheorigintothepointthatrepresents),(yxz(Sec.1.1)inthecomplexplane.Infact,weoftenrefertozasthepointzorthevectorz.InFig.1-2,thenumberiyxz+=andi+−2aredisplayedgraphicallyasbothpointsandradiusvectors.Fig.1-2Accordingtothedefinitionofthesumoftwocomplexnumbersand,111iyxz+=222iyxz+=21zz+maybeobtainedvectoriallyasshowninFig.1-3.Thedifference)(2121zzzz−+=−correspondstothesu