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1、EXERCISESofCOMPLEXANALYSISⅠ.ClozeTests1.Themodulusofthenumberis2.Themodulusofthenumberis53.4.Themainargumentandthemodulusofthenumberare5.Thesquarerootsof1+are.6.Thedefinitionofis.7.Log=.8.Ifdenotesthecirclecenteredatpositivelyorientedandisapositiveinteger，then.9.Th
2、eintegralofthefunctiononis.10.Thesolutionsoftheequationare.11.12.13.14.015.ThesolutionsoftheequationareⅡ.TrueorFalseQuestions1.Ifafunctionisdifferentiableatapoint，thenitiscontinuousat.（T）2.Afunctionisdifferentiableatapointifandonlyifwhoserealandimaginarypartsaredif
3、ferentiableatandtheCauchyRiemannconditionsholdthere.（）3.Ifafunctionisanalyticatapoint,thenitisdifferentiableat.（）4.Afunctionisanalyticatapointifandonlyifwhoserealandimaginarypartsaredifferentiableat.（）5..（T）6..（）7..（）8..9.Theexponentialfunctionisperiodic.（T）10.Ifaf
4、unctionisdifferentiableatapoint,thenitisanalyticat.（）11.Ifisaharmonicconjugateofinsomedomain,thenisaharmonicconjugateofthere.（）12.Thelogarithmicfunctionisentire.（）Ⅲ.Computations1.Evaluatetheintegral,whereisthepositivelyorientedcircle.Solution:Thecirclemayberepresen
5、tedparametricallyas.Consequently,;Sinceand,Thismeansthat2.Evaluatetheintegral,whereisdenotedthesemicircularpathfromthepointtothepoint.3.Find4.FindthevalueofSolution:Sinceisaninteriortoandareoutofthecircle,let,wecangetthatConsidingthatisannlyticwithinandonthecircle,
6、wehave5.Findthevalueof6.FindthevalueofSolution:Itisclearthatisaninteriorto.Let,sinceisentire,wehave.7.Given，where，find.Solution:Let,andweknowthatisanentirefunction.ThuswegetaccordingtotheCauchyintegralformula.Thus,.Since,wehave.8.Given，where，find.Ⅳ.Verifications1.A
7、ssumeafunctionisanalyticthroughoutagivendomainDanditsmodulusisconstantonD.ShowthatthefunctionmustbeconstantonD.Proof:ConsideringisconstantonD,wedivideintoand.When,obviously,wegetthatonD.When,thefactthattellsusthatisneverzeroonD.HenceforallzinD,Andwegetthatisanalyti
8、conDbytheassumptionthatisanalyticonD.ThusitiseasytoshowthatmustbeconstantonD.2.Showthatthefunctionisentire.Proof:Thecomponentfunctionsareand.Beca
