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1、ConciseComplexAnalysisSolutionofExerciseProblemsAiShuXueMarch9,20081Contents1Calculus32CauchyIntegralTheoremandCauchyIntegralFormula143TheoryofSeriesofWeierstrass274RiemannMappingTheorem525Di?erentialGeometryandPicard'sTheorem536AFirstTasteofFunctionTheoryofSeveralComplexVar
2、iables537EllipticFunctions538TheRiemann3-FunctionandThePrimeNumberTheory532ThisisasolutionmanualofselectedexerciseproblemsfromConciseComplexAnalysis,SecondEdition,byGongSheng(WorldScientiˉc,2007).ThisversionsolvestheexerciseproblemsinChapter1-3,exceptthefollowing:Chapter1pro
3、blem37-42;Chapter2problem47,49;Chapter3problem15(xi).1Calculus1.Proof.See,forexample,Munkres[4],x38.2.(1)pProof.j2ij=2,arg(2i)=?.j1?ij=2,arg(1?i)=7?.j3+4ij=5,arg(3+4i)=arctan4?0:927324?￠3(Matlabcommand:atan(4=3)).j?5+12ij=13,arg(?5+12i)=arccos?5?1:9656(Matlabcommand:13acos(?
4、5=13)).(2)310862?3i514nnn+1nProof.(1+3i)=?26?18i.=+i.=?i.(1+i)+(1?i)=22cos?.4?3i554+i17174(3)pˉˉpProof.j?3i(2?i)(3+2i)(1+i)j=3130.ˉˉ(4?3i)(2?i)ˉˉ=p5￠p5=5.(1+i)(1+3i)2￠1023.pp?11?iμi4Proof.Letμ=arctan,?=arctan.Then5?i=26e,1+i=2e4and(5?i)(1+i)=p5239pi(??4μ)4??i?6762e4.Meanwhil
5、e(5?i)(1+i)=956?4i=6762e.Sowemusthave?4μ=??+2k?,4k2Z,i.e.?=4arctan1?arctan1+2k?,k2Z.Since4arctan1?arctan1+2?>0??+2?=7?>?452395239244and4arctan1?arctan1?2?<4￠??2?=0
6、z1x2+y2x2+y21?z(1?x)2+y2(1?x)2+y2z2+13232x+xy+x+i(?y?xy+y)(x2?y2+1)2+4x2y2.5.Proof.a=1,b=?+iˉ,c=°+i±.So￠=b2?4ac=(?2?ˉ2?4°)+i(2?ˉ?4±)andp?(?+iˉ)§￠z=:26.222Proof.Denoteargzbyμ,thenz+r=reiμ+r=r(eiμ+e?iμ)=2rcosμ=2Rez,andz?r=zreiμz2reiμ?r=r(eiμ?e?iμ)=2irsinμ=2iImz.reiμ7.3Proof.If
7、a=reiμ1andb=reiμ2,then12ˉˉˉˉˉa?bˉˉr?rei(μ2?μ1)ˉˉˉ=ˉ12ˉ:ˉ1?ab1ˉˉ1?r1r2ei(μ2?μ1)ˉDenoteμ?μbyμ,wecanreducetheproblemtocomparingjr?reiμj2andj1?rreiμj2.Note211212jr?reiμj2=(r?rcosμ)2+r2sin2μ=r2?2rrcosμ+r2121221122andj1?rreiμj2=(1?rrcosμ)2+r2r2sin2μ=1?2rrcosμ+r2r2:1212121212ˉˉSoj1
8、?rreiμj2?jr?reiμj2=(r2?1)(r2?1).Thisobservationshowsˉa?bˉ121212ˉ1?ab1ˉ=1ifa
