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1、SOLUTIONS/HINTSTOTHEEXERCISESFROMCOMPLEXANALYSISBYSTEINANDSHAKARCHIROBERTC.RHOADESAbstract.ThiscontainsthesolutionsorhintstomanyoftheexercisesfromtheComplexAnalysisbookbyEliasSteinandRamiShakarchi.IworkedtheseproblemsduringtheSpringof2006whileIwastakingaComplexAnalysiscoursetaugh
2、tbyAndreasSeegerattheUniversityofWisconsin-Madison.Iamgratefultohimforhiswonderfullecturesandhelpfulconversationsaboutsomeoftheproblemsdiscussedbelow.Contents1.Chapter1.PreliminariestoComplexAnalysis22.Chapter2.Cauchy’sTheoremandItsApplications83.Chapter3.MeromorphicFunctionsandt
3、heLogarithm94.Chapter4.TheFourierTransform105.Chapter5:EntireFunctions116.Chapter6.TheGammaandZetaFunctions137.Chapter7:TheZetaFunctionandPrimeNumberTheorem178.Chapter8:ConformalMappings209.Chapter9:AnIntroductiontoEllipticFunctions2310.Chapter10:ApplicationsofThetaFunctions25Dat
4、e:September5,2006.TheauthoristhankfulforanNSFgraduateresearchfellowshipandaNationalPhysicalScienceConsortiumgraduatefellowshipsupportedbytheNSA.12ROBERTC.RHOADES1.Chapter1.PreliminariestoComplexAnalysisExercise1.Describegeometricallythesetsofpointszinthecomplexplanedefinedbythefol
5、lowingrelations:(1)
6、z−z1
7、=
8、z−z2
9、wherez1,z2∈C.(2)1/z=z.(3)Re(z)=3.(4)Re(z)>c,(resp.,≥c)wherec∈R.(5)Re(az+b)>0wherea,b∈C.(6)
10、z
11、=Re(z)+1.(7)Im(z)=cwithc∈R.Solution1.(1)Itisthelineinthecomplexplaneconsistingofallpointsthatareanequaldistancefrombothz1andz2.Equivalentlytheperpendicular
12、bisectorofthesegmentbetweenz1andz2inthecomplexplane.(2)Itistheunitcircle.(3)Itisthelinewhereallthenumbersonthelinehaverealpartequalto3.(4)Inthefirstcaseitistheopenhalfplanewithallnumberswithrealpartgreaterthanc.Inthesecondcaseitistheclosedhalfplanewiththesamecondition.(5)(6)Calcul
13、ate
14、z
15、2=x2+y2=(x+1)2=x2+2x+1.Soweareleftwithy2=2x+1.Thusthecomplexnumbersdefinedbythisrelationisaparabolaopeningtothe“right”.(7)Thisisaline.Exercise2.Leth·,·idenotetheusualinnerproductinR2.Inotherwords,ifZ=(z,y)and11W=(x2,y2),thenhZ,Wi=x1x2+y1y2.Similarly,wemaydefineaHermitianinner
16、product(·,·)inCby(z,w)=zw.ThetermHermiti
