DepartmentofMathematicsandStatisticsIndianInstituteofTechnologyKanpur

MSO202AAssignment3IntroductionToComplexAnalysis

The problems marked (T) need an explicit discussion in the tutorial class. Other problems areforenhancedpractice.

1.Evaluate

(a) ,C

z z dz whereC isthecounterclockwiseorientedsemicircularpartof thecircle 2z lying in

thesecondandthirdquadrants.

(T)(b) ,C

zz dz

z where C is the clockwise oriented boundary of the part of the annulus

2 4z lyinginthethirdandfourthquadrants.

(c) ( 2 ) ,n

C

z a dz whereCisthesemicircle 2 , 0 arg( 2 )z a R z a .

2. Evaluatetheintegral1

C

dzz ,ineachofthefollowingcases:

(a) Cisthecounterclockwiseorientedsemicircularpartofthecircle 1z intheupperhalfplane

and z isdefinedsothat 1 1. (T)(b)Cisthecounterclockwiseorientedsemicircularpartofthecircle 1z inthelowerhalfplane

and z isdefinedsothat 1 1. (c)Cistheclockwiseorientedcircle 1z and z isdefinedsothat 1 1.

(d)Cisthecounterclockwiseorientedcircle 1z and z isdefinedsothat 1 .i

3. Withoutactuallyevaluatingtheintegral,provethat

(T)(a)2

1,

1 3C

dzz

whereC is thearcof the circle 2z fromz=2 to z=2i lying in the first

quadrant.

(b) 2 2( 1) ( 1),C

z dz R R whereCisthesemicircleofradiusR>1withcenterattheorigin.

4. (T)DoesCauchyTheoremholdseparatelyfortherealorimaginarypartofananalyticfunctionf(z)?Whyorwhynot?

5. Aboutthepointz=0,determinetheTaylorseriesforeachofthefollowingfunctions:

(T) 2( ) 2 ( ) ( 3 2)i z i ii Log z z 6. Abouttheindicatedpointz=z0,determinetheTaylorseriesanditsregionofconvergenceforeachof

thefollowingfunctions.Ineachcase,doestheTaylorseriesnecessarilysumsuptothefunctionateverypointofitsregionofconvergence?

0

1( ) , 1

1i z

z

(T) 0( ) cosh ,ii z z i (T) 0( ) , 1iii Log z z i

2

7. (T)Evaluate the integral2( 1)C

dz

z z , for all possible choices of the contour C that does not pass

throughanyofthepointsz=0, i .8. (T)UseCauchyTheoremformultiplyconnecteddomainsandCauchyIntegralFormulatoevaluate

theintegral

2cos

( 1) 2C

zdz

z z

,C:thecircle 3z orientedcounterclockwise.

9. Evaluate

(T)(a)2

4( 1)

z

C

edz

z z ,C:thecircle 2z orientedclockwise

(b)2

sin

( ) nC

zdz

z ,C:thecircle 1z orientedcounterclockwise

10. Ifuisaharmonicfunctionin 0 ,z R and r R showthat

2

0

1(0) ( ) .

2iu u re d