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LectureNote
2301308FunctionsofAComplexVariable
PhichetChaoha
DepartmentofMathematics,FacultyofScience,Chulalongkorn
University,Bangkok10330,Thailand
Contents
Chapter1.ComplexNumbersandFunctions 5
Chapter2.TopologyofC:AFastGlimpse 7
Chapter3.DierentiabilityandAnalyticity 13
Chapter4.ElementaryFunctions 15
Chapter5.LineIntegrals 17
Chapter6.CauchyIntegralTheoremandApplications 21
Chapter7.SequencesandSeriesofComplexFunctions 27
Chapter8.Singularity 35
Bibliography
39
3
CHAPTER1
ComplexNumbersandFunctions
=C−{0}.NoticethatRC
(bylettingb=0)andthefunction:C!R
2
givenby(a+bi)=(a,b)isclearly
abijection.WeusuallyusethisbijectiontoidentifyCwithR
2
(i.e.,a+bi=(a,b))
whileRcanbeviewedastheX-axisofR
2
.Inthisnote,wewillregarda+bias
thestandardformand(a,b)asthevectorformofacomplexnumber.
SomeNotations:Foracomplexnumberz=a+bi,
•therealpartofzisRe(z)=a,
•theimaginarypart
of
zisIm(z)=b,
•theconjugateofzisz=a
−bi,
•themodulusofzis|z|=
a
2
+b
2
,
•anargumentofz6=0isananglebetweenthevectors(0,1)and(a,b)
(viewinginR
2
)measuredinthecounter-clockwisedirection.Noticethat
arg(z)ismultivalued.Weusuallycalltheargumentthatliesintheinterval
(−,]theprincipalargumentofzanddenoteitbyArg(z).
OtherFormsofComplexNumbers:Foracomplexnumberz=a+bi,
letr=|z|and=arg(z).[Eulerformula:e
i
=cos+isin.]
•Thepolarformofzisz=r(cos+isin).
•Theexponentialformofzisz=re
i
.
Noticethatbothpolarformandexponentialformofacomplexnumberzisnot
uniqueandwealwayshave|cos+isin|=|e
i
|=1.
ComplexAlgebra:Fortwocomplexnumbersz=a+biandw=c+di,
wedefine
p
z+w=(a+c)+(b+d)i
and
z·w=(ac−bd)+(ad+bc)i.
a
2
+b
2
iasthemultiplicativeinverseofa+bi6=0.Asusual,wewill
denotetheadditiveinverseandthemultiplicativeinverse(ifexists)ofzby−zand
z
−1
(or
1
z
)respectively.
Intermsofpolarandexponentialforms,ifz
1
=r
1
(cos
1
+isin
1
)=r
1
e
i
1
andz
2
=r
2
(cos
2
+isin
2
)=r
2
e
i
2
,onecanshowthat
a
2
+b
2
−
b
z
1
z
2
=r
1
r
2
(cos(
1
+
2
)+isin(
1
+
2
))=r
1
r
2
e
in(
1
+
2
)
5
Acomplexnumberisanexpressionoftheforma+bi(ora+ib)where
a,b2Randiisasymbolsatisfyingtherelationi
2
=−1.Thesetofallcomplex
numberswillbedenotedbyCandwealsoletC
Itisstraightforwardtoverifythat(C,+,·)isafieldwith0astheadditive
identity,1asthemultiplicativeidentity,−a−biastheadditiveinverseofa+bi
and
a
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