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UPSCCivilServicesMain2002-Mathematics

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ComplexAnalysisSunderLalRetired Professor of MathematicsPanjab University

ChandigarhJanuary30,2010Question1(a)SupposethatfandgaretwoanalyticfunctionsonthesetCofallcomplex

numberswithf(n1

)=g(n1

)forn=1,2,3,...,thenshowthatf(z)=g(z)forallzC.

Solution.LetG(z)=G(z)LetG(z)0for=

z

Cwhichf(z)wouldg(z),provethentheresult.

G(n1

)=0forn=1,2,....WeshallshowthatR,clearlyR>0.n=0

Weashallnznbenowtheprovepowerthatseriesan

of=G(z)0forwitheverycentern.0andradiusofconvergenceIfan

=0forsomen,letak

bethefirstnon-zerocoefficient.Then

G(z)=zk(ak

z+...)=zkH(z)

ClearlyH(z)isanalyticin|z|

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H(0)|

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)=0asz1

K0

K

1,andK0

K1

containsasequenceofpointsyn

suchthatyn

z1

andG(yn)=0,wecanproveasbeforethatG(z)0inK1

.Proceedinginthisway,innstepswegetG(z)0inKn

,orG(z)=0.SincezisanarbitrarypointofC,weget

G(z)0inC.

1

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z

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Question2(a)Showthatwhen0

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whichisthedesiredLaurentseriesexpansion.Question2(b)Establishbycontourintegration0

3cos(ax)x2+12ea,wherea0Solution.LetIbethegivenintegral.Putax=t,sothatI=dx=

costdt0

t2

a2

a0

t2cost+a2dtWeshallnowprovethat+1=a0

cost

t2+a22aea,whichwillshowthatI=2

eaasrequired.Clearly

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dt=Weconsiderx2cosx+thea2integralisthereal

partf(z)dzofx2whereeix+a2.f(z)ingof=thez2eizline+a2

joiningand(R,0)isthecontourand(R,0)consist-and,whichisthearcofthecircleofradiusRandcenter(0,0)lyingintheupperhalfplane.(R,0)(0,0)(R,0)2

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Clearlyon,ifweputz=Rei,then0and

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z2eiz+dza2

=

0

RieieiReiR2e2i+a2d

0

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R2e2iRieieiRei+a2dBut|eiRei|=|eiRcoseRsin|=eRsin1assin0for0.|z2+a2||z|2a2=

R2a2.Therefore

z2eiz+dza2

0

RR2a2RR2a2

Henced=NowButthez2eizz2only+eizdz+a2

dza2pole=02i(suminastheRupper.ofresidueshalfplaneatpolesisz=insideia,(a).>0)andtheresidueatz=iaisei(ia)2ia

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=ea2ia.ThusR

lim

eizdz

z2+a2

eixdxx2+a2

ea2iaeaa

=

==2i=

cosxdxx2+a2=eaa

,

sinxdxx2+a2=0

=

0

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cosxdxx2+a2=ea2a

cosx=cos(x)

Thiscompletestheproof.3