Solutions Manual for FunctionsofOneComplexVariableI,SecondEdition1 © Copyright by Andreas Kleefeld, 2009 All Rights Reserved2 1byJohnB.Conway 2LastupdatedonJanuary,7th2013 PREFACE MostoftheexercisesIsolvedwereassignedhomeworksinthegraduatecoursesMath713andMath714 ”ComplexAnalysisIandII”attheUniversityofWisconsin–MilwaukeetaughtbyProfessorDashanFan inFall2008andSpring2009. ThesolutionsmanualisintentedforallstudentstakingagraduatelevelComplexAnalysiscourse. Students canchecktheiranswerstohomeworkproblemsassignedfromtheexcellentbook“FunctionsofOneCom- plexVariableI”,SecondEditionbyJohnB.Conway. Furthermorestudentscanprepareforquizzes,tests, examsandfinalexamsbysolvingadditionalexercisesandchecktheirresults. Maybestudentsevenstudy forpreliminaryexamsfortheirdoctoralstudies. However, I have to warn you not to copy straight of this book and turn in your homework, because this wouldviolatethepurposeofhomeworks. Ofcourse,thatisuptoyou. Istronglyencourageyoutosendmesolutionsthatarestillmissingtokleefeld@tu-cottbus.de(LATEXpreferred butnotmandatory)inordertocompletethissolutionsmanual. Thinkaboutthecontributionyouwillgive tootherstudents. If you find typing errors or mathematical mistakes pop an email to [email protected]. The recent versionofthissolutionsmanualcanbefoundat http://www.math.tu-cottbus.de/INSTITUT/kleefeld/Files/Solution.html. Thegoalofthisprojectistogivesolutionstoallofthe452exercises. CONTRIBUTION Ithank(withoutspecialorder) ChristopherT.Alvin MartinJ.Michael DavidPerkins forcontributionstothisbook. 2 Contents 1 TheComplexNumberSystem 1 1.1 Therealnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Thefieldofcomplexnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Thecomplexplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Polarrepresentationandrootsofcomplexnumbers . . . . . . . . . . . . . . . . . . . . . 5 1.5 Linesandhalfplanesinthecomplexplane . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Theextendedplaneanditssphericalrepresentation . . . . . . . . . . . . . . . . . . . . . 7 2 MetricSpacesandtheTopologyofC 9 2.1 Definitionsandexamplesofmetricspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Sequencesandcompleteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Uniformconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 ElementaryPropertiesandExamplesofAnalyticFunctions 21 3.1 Powerseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Analyticfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Analyticfunctionsasmappings. Möbiustransformations . . . . . . . . . . . . . . . . . . 31 4 ComplexIntegration 42 4.1 Riemann-Stieltjesintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Powerseriesrepresentationofanalyticfunctions. . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Zerosofananalyticfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Theindexofaclosedcurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Cauchy’sTheoremandIntegralFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.6 ThehomotopicversionofCauchy’sTheoremandsimpleconnectivity . . . . . . . . . . . 63 4.7 Countingzeros;theOpenMappingTheorem . . . . . . . . . . . . . . . . . . . . . . . . 66 4.8 Goursat’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 Singularities 68 5.1 Classificationofsingularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.2 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3 TheArgumentPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3 6 TheMaximumModulusTheorem 84 6.1 TheMaximumPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 Schwarz’sLemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.3 ConvexfunctionsandHadamard’sThreeCirclesTheorem . . . . . . . . . . . . . . . . . 88 6.4 ThePhragmén-LindelöfTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7 CompactnessandConvergenceintheSpaceofAnalyticFunctions 93 7.1 ThespaceofcontinuousfunctionsC(G,Ω) . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 Spacesofanalyticfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.3 Spacesofmeromorphicfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7.4 TheRiemannMappingTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.5 TheWeierstrassFactorizationTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.6 Factorizationofthesinefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.7 Thegammafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.8 TheRiemannzetafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 8 Runge’sTheorem 123 8.1 Runge’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.2 Simpleconnectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.3 Mittag-Leffler’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9 AnalyticContinuationandRiemannSurfaces 130 9.1 SchwarzReflectionPrinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 9.2 AnalyticContinuationAlongaPath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 9.3 MonodromyTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.4 TopologicalSpacesandNeighborhoodSystems . . . . . . . . . . . . . . . . . . . . . . . 132 9.5 TheSheafofGermsofAnalyticFunctionsonanOpenSet . . . . . . . . . . . . . . . . . 133 9.6 AnalyticManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9.7 Coveringspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10 HarmonicFunctions 137 10.1 Basicpropertiesofharmonicfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 10.2 Harmonicfunctionsonadisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.3 Subharmonicandsuperharmonicfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . 144 10.4 TheDirichletProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 10.5 Green’sFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 11 EntireFunctions 151 11.1 Jensen’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 11.2 Thegenusandorderofanentirefunction . . . . . . . . . . . . . . . . . . . . . . . . . . 153 11.3 HadamardFactorizationTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 12 TheRangeofanAnalyticFunction 161 12.1 Bloch’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 12.2 TheLittlePicardTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 12.3 Schottky’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 12.4 TheGreatPicardTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4 Chapter 1 The Complex Number System 1.1 The real numbers Noexercisesareassignedinthissection. 1.2 The field of complex numbers Exercise1. Findtherealandimaginarypartsofthefollowing: 1 z a 3+5i 1+i√3 3 ; − (a R);z3; ; − ; z z+a ∈ 7i+1  2    1 i√3 6 1+i n − − ;in; for 2 n 8.  2  √2 ! ≤ ≤ Solution. Letz= x+iy. Then  a) 1 x Re(z) Re = = z! x2+y2 z2 | | 1 y Im(z) Im = = z! −x2+y2 − z2 | | b) z a x2+y2 a2 z2 a2 Re − = − = | | − z+a x2+y2+2ax+a2 z2+2aRe(z)+a2 (cid:18) (cid:19) | | z a 2ya 2Im(z)a Im − = = z+a x2+y2+2xa+a2 z2+2aRe(z)+a2 (cid:18) (cid:19) | | c) Re z3 = x3 3xy2 =Re(z)3 3Re(z)Im(z)2 − − (cid:16) (cid:17) Im z3 = 3x2y y3 =3Re(z)2Im(z) Im(z)3 − − (cid:16) (cid:17) 1 d) 3+5i 19 Re = 7i+1! 25 3+5i 8 Im = 7i+1! −25 e) 1+i√3 3 Re − = 1  2    1+i√33 Im − = 0  2      f) 6 1 i√3 Re − − = 1  2    1 i√36 Im − − = 0  2      g) 0, nisodd Re(in) = 1, n 4k:k Z  1, n∈{2+4k∈:k} Z 0−, ni∈s{even ∈ } Im(in) = 1, n 1+4k:k Z  1, n∈{3+4k:k∈Z} − ∈{ ∈ } 2 h) 0, n=2  √2, n=3 Re 1√+2i!n! = −−0−,1√22,2, nnn===654 11√2,,2, nnn===827 √2, n=3 Im 1√+2i!n! = −02,1√2,2, nnn===456 −0−,√22, nn==87 Exercise2. Findtheabsolutevalueandconjugateofeachofthefollowing: 3 i i 2+i; 3;(2+i)(4+3i); − ; ;(1+i)6;i17. − − √2+3i i+3 Solution. Itiseasytocalculate: a) z= 2+i, z = √5, z¯= 2 i − | | − − b) z= 3, z =3, z¯= 3 − | | − c) z=(2+i)(4+3i)=5+10i, z =5√5, z¯=5 10i | | − d 3 i 1 3+i z= − , z = √110, z¯= √2+3i | | 11 √2 3i − e) i 1 3 1 1 3 z= = + i, z = √10, z¯= i i+3 10 10 | | 10 10 − 10 f) z=(1+i)6 = 8i, z =8, z¯=8i − | | g) i17 =i, z =1, z¯= i | | − Exercise3. Showthatzisarealnumberifandonlyifz=z¯. 3 Solution. Letz= x+iy. :Ifzisarealnumber,thenz= x(y=0). Thisimpliesz¯= xandthereforez=z¯. ⇒:Ifz=z¯,thenwemusthavex+iy= x iyforallx,y R. Thisimpliesy= ywhichistrueify=0and t⇔hereforez= x. Thismeansthatzisarea−lnumber. ∈ − Exercise4. Ifzandwarecomplexnumbers,provethefollowingequations: z+w2 = z2+2Re(zw¯)+ w2. | | | | | | z w2 = z2 2Re(zw¯)+ w2. | − | | | − | | z+w2 + z w2 =2 z2+ w2 . | | | − | | | | | (cid:16) (cid:17) Solution. Wecaneasilyverifythatz¯¯=z. Thus z+w2 = (z+w)(z+w)=(z+w)(z¯+w¯)=zz¯+zw¯ +wz¯+ww¯ | | = z2+ w2+zw¯ +z¯w= z2+ w2+zw¯ +z¯w¯¯ | | | | | | | | zw¯ +zw¯ = z2+ w2+zw¯ +zw¯ = z2+ w2+2 | | | | | | | | 2 = z2+ w2+2Re(zw¯)= z2+2Re(zw¯)+ w2. | | | | | | | | z w2 = (z w)(z w)=(z w)(z¯ w¯)=zz¯ zw¯ wz¯+ww¯ | − | − − − − − − = z2+ w2 zw¯ z¯w= z2+ w2 zw¯ z¯w¯¯ | | | | − − | | | | − − zw¯ +zw¯ = z2+ w2 zw¯ zw¯ = z2+ w2 2 | | | | − − | | | | − 2 = z2+ w2 2Re(zw¯)= z2 2Re(zw¯)+ w2. | | | | − | | − | | z+w2+ z w2 = z2+Re(zw¯)+ w2+ z2 Re(zw¯)+ w2 | | | − | | | | | | | − | | = z2+ w2+ z2+ w2 =2z2+2w2 =2 z2+ w2 . | | | | | | | | | | | | | | | | (cid:16) (cid:17) Exercise5. Useinductiontoprovethatforz=z +...+z ;w=w w ...w : 1 n 1 2 n w = w ... w ;z¯=z¯ +...+z¯ ;w¯ =w¯ ...w¯ . 1 n 1 n 1 n | | | | | | Solution. Notavailable. Exercise6. LetR(z)bearationalfunctionofz. ShowthatR(z)=R(z¯)ifallthecoefficientsinR(z)arereal. Solution. LetR(z)bearationalfunctionofz,thatis a zn+a zn 1+...a R(z)= b nzm+bn−1zm− 1+...b0 m m 1 − 0 − wheren,marenonnegativeintegers. LetallcoefficientsofR(z)bereal,thatis a ,a ,...,a ,b ,b ,...,b R. 0 1 n 0 1 m ∈ Then a zn+a zn 1+...a a zn+a zn 1+...a R(z) = bmnzm+bnm−−11zm−−1+...b00 = bmnzm+bmn−−11zm−−1+...b00 a zn+a zn 1+...a a z¯n+a z¯n 1+...a = bmnzm+bmn−−11zm−−1+...b00 = bmnz¯m+bmn−−11z¯m−−1+...b00 =R(z¯). 4 1.3 The complex plane Exercise1. Prove(3.4)andgivenecessaryandsufficientconditionsforequality. Solution. Letzandwbecomplexnumbers. Then z w = z w+w w || |−| || || − |−| || z w + w w ≤ || − | | |−| || = z w || − || = z w | − | Noticethat z and w isthedistancefromzandw, respectively, totheoriginwhile z w isthedistance | | | | | − | betweenzandw. Consideringtheconstructionoftheimpliedtriangle,inordertoguaranteeequality,itis necessaryandsufficientthat z w = z w || |−| || | − | (z w)2 = z w2 ⇐⇒ | |−| | | − | (z w)2 = z2 2Re(zw¯)+ w2 ⇐⇒ | |−| | | | − | | z2 2z w + w2 = z2 2Re(zw¯)+ w2 ⇐⇒ | | − | || | | | | | − | | z w =Re(zw¯) ⇐⇒ | || | zw¯ =Re(zw¯) ⇐⇒ | | Equivalently, this is zw¯ 0. Multiplying this by w, we get zw¯ w = w2 z 0 if w , 0. If ≥ w · w | | · w ≥ t= z = 1 w2 z. Thent 0andz=tw. w w2 ·| | · w ≥ (cid:16)| | (cid:17) Exercise2. Showthatequalityoccursin(3.3)ifandonlyifz /z 0foranyintegerskandl,1 k,l n, k l ≥ ≤ ≤ forwhichz ,0. l Solution. Notavailable. Exercise3. Leta Randc>0befixed. Describethesetofpointszsatisfying ∈ z a z+a =2c | − |−| | foreverypossiblechoiceofaandc. Nowletabeanycomplexnumberand,usingarotationoftheplane, describethelocusofpointssatisfyingtheaboveequation. Solution. Notavailable. 1.4 Polar representation and roots of complex numbers Exercise1. Findthesixthrootsofunity. Solution. Start with z6 = 1 and z = rcis(θ), therefore r6cis(6θ) = 1. Hence r = 1 and θ = 2kπ with k 6 ∈ 3, 2, 1,0,1,2 . Thefollowingtablegivesalistofprinciplevaluesofargumentsandthecorresponding {v−alue−of−therootof}theequationz6 =1. θ =0 z =1 0 0 θ = π z =cis(π) 1 3 1 3 θ = 2π z =cis(2π) 2 3 2 3 θ =π z =cis(π)= 1 3 3 θ = 2π z =cis( 2π) − 4 −3 4 −3 θ = π z =cis( π) 5 −3 5 −3 5 Exercise2. Calculatethefollowing: a) thesquarerootsofi b) thecuberootsofi c) thesquarerootsof √3+3i Solution. c)Thesquarerootsof √3+3i. Letz= √3+3i. Thenr = z = √3 2+32 = √12andα=tan 1 3 = π. So,the2distinctrootsofz aregivenby √2r cosα+2kπ +| |isinqα+(cid:16)2kπ(cid:17)wherek=0,1. Specifically−, (cid:18)√3(cid:19) 3 n n (cid:16) (cid:17) π +2kπ π +2kπ √z= √412 cos 3 +isin 3 . 2 2 ! Therefore,thesquarerootsofz,z ,aregivenby k z = √412 cosπ +isinπ = √412 √3 + 1i 0 6 6 2 2 (cid:16) (cid:17) (cid:18) (cid:19) z = √412 cos7π +isin7π = √412 √3 1i . 1 (cid:16) 6 6 (cid:17) (cid:18)− 2 − 2 (cid:19) So,inrectangularform,thesecondrootsofzaregivenby √4108, √412 and √4108, √412 . (cid:18) 2 2 (cid:19) (cid:18)− 2 − 2 (cid:19) Exercise3. Aprimitiventhrootofunityisacomplexnumberasuchthat1,a,a2,...,an 1 aredistinctnth − rootsofunity.Showthatifaandbareprimitiventhandmthrootsofunity,respectively,thenabisakthroot ofunityforsomeintegerk. Whatisthesmallestvalueofk? Whatcanbesaidifaandbarenonprimitive rootsofunity? Solution. Notavailable. Exercise4. Usethebinomialequation n n (a+b)n = an kbk, k ! − Xk=0 where n n! = , k ! k!(n k)! − andcomparetherealandimaginarypartsofeachsideofdeMoivre’sformulatoobtaintheformulas: n n cosnθ = cosnθ cosn 2θsin2θ+ cosn 4θsin4θ ... − 2 ! − 4 ! − − n n sinnθ = cosn 1θsinθ cosn 3θsin3θ+... 1 ! − − 3 ! − Solution. Notavailable. Exercise5. Letz=cis2π foranintegern 2. Showthat1+z+...+zn 1 =0. n ≥ − 6