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1、ComplexNumbersandFunctions______________________________________________________________________________________________NaturalisthemostfertilesourceofMathematicalDiscoveries-JeanBaptisteJosephFourierTheComplexNumberSystemDefinition:Acomplexnumberzisanumberoftheformzai=+b,wherethesymb
2、oli=-1iscalledimaginaryunitandabR,.Îaiscalledtherealpartandbtheimaginarypartofz,writtenaz=Reandbz=Im.Withthisnotation,wehavezz=+ReizIm.ThesetofallcomplexnumbersisdenotedbyCai=+{}b,.abRÎIfb=0,thenzaia=+=0,isarealnumber.Alsoifa=0,thenzi=+=0,bibisaimaginarynumber;inthiscase,ziscalledpure
3、imaginarynumber.Letaib+andcid+becomplexnumbers,withabcdR,,,Î.1.Equalityaibcid+=+ifandonlyifac==andbd.Note:Inparticular,wehavezai=+=b0ifandonlyifab==00and.2.FundamentalAlgebraicPropertiesofComplexNumbers(i).Addition()aibcidacibd+++=+++()()().(ii).Subtraction()aibcidacibd+-+=-+-()()().(
4、iii).Multiplication()a++=-++ibcid()()()acbdiadbc.Remark(a).Byusingthemultiplicationformula,onedefinesthenonnegativeintegralpowerofacomplexnumberzas1232nn-1zzzz===,,,zzzz!,.zzz=0Furtherforz¹0,wedefinethezeropowerofzis1;thatis,z=1.(b).Bydefinition,wehave234i=-1,ii=-,i=1.(iv).DivisionIfc
5、id+¹0,thenaib+æacbd+öæbc-adö=ç22÷+iç22÷.cid+ècd+øècd+øRemark(a).Observethatifaib+=1,thenwehave1æcöæ-dö=ç÷+iç÷.è22øè22øcid+cd+cd+(b).Foranynonzerocomplexnumberz,wedefine1-1z=,z-1whereziscalledthereciprocalofz.(c).Foranynonzerocomplexnumberz,wenowdefinethenegativeintegralpowerofacomplex
6、numberzas1--12-1-1-3-2--1nn-+1-1z==,,,zzzzz=z!,.zzz=z1-1-2-3-4(d).i==-i,i=-1,ii=,i=1.i3.MorePropertiesofAdditionandMultiplicationForanycomplexnumberszzz,,andz,123(i).CommutativeLawsofAdditionandMultiplication:zzzz+=+;1221zz=zz.1221(ii)AssociativeLawsofAdditionandMultiplication:zzzzzz+
7、+=++()();123123zzz()().=zzz123123(iii).DistributiveLaw:zzz()+=+zzzz.1231213(iv).AdditiveandMultiplicativeidentities:zz+=+=00z;zz×=×=11z.(v).zzzz+-=-+=()()0.ComplexConjugateandTheirPropertiesDefinition:Let.z=a+ibÎC,a,bÎRThecomplexconjugate,orbrieflyconjugate,ofzisdefinedbyzai=-b.Forany
8、compl
