Table Of ContentSolutions 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√33
Im − = 0
2
f)
6
1 i√3
Re − − = 1
2
1 i√36
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
