BITS Pilani Pilani Campus

MATH F112 (Mathematics-II)

Complex Analysis

BITS Pilani Pilani Campus

Lecture 30-34

Integrals

Dr Trilok Mathur,

Assistant Professor,

Department of Mathematics

BITS Pilani, Pilani Campus

. of functions valued-real are and

where, variable real a of function valued

complex a be Let 1

tvu

t

tvitutw

t vu

tvitutwdt

dw

at exists & sderivative the of each

provided , Then

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.dt

dwztwz

dt

d

z

00

02

then constant,complex a is If

.3 00

0

tztzeze

dt

d

true. NOT is sderivative

for Theorem ValueMean 4

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ba,

vu

bt, atvi t u t wi

on

continuous are and i.e.,continuous

be

:thatSuppose

)()()()(

. in exists btat wii )()(

ab

awbw cw

a, bc

)()()(

)(

that such

in any exist NOTmay there Then

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πt, e tw it 20)( Let

:Example

π, t tw

π, ttw

ei tw it

200)(

201)(

)(

all for

all for

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0

)0()2( 02

ee wπw i.πi But

plane.complex the in true

NOT in derivative for MVT

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.

over functions valued-real :

. variable-real a of function valued

complex a be Let

b ta

t, vtu

t

tvitutw

)()(

)()()(

exist. right the on integrals individual

the where

as defined is of integral definite Then

b

a

b

a

b

adttvidttudttw

tw

,)()()(

)(

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b

a

b

a

b

a

b

a

dttwdttw

dttwdttw

.)(Im)(Im&

,)(Re)(Re

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1

0

21

0

2211 dttitdtti

:Example

i

dttidtt

3

2

211

0

1

0

2

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b

a

b

adttwdttw

ba,

tw

)()(

.

)(

Then on integrable function

valued-complex a be Let

:Property

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. parameter

real a of functions continuous are

if curve a

be to said is planecomplex the in

points of set A : (1)

sDefinition

t

ty, ytxx

C

yx,z Curve

)()(

)(

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Or

writeWe

.tyitx tC:z

tyy,txC:x

)()()(

)()(

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arc. an is ,

i.e. curve, the ofarc an called is curve a

of points twoany between portion The

: (2)

,)()(: btat i y txC

Arc

curve. the ofarc an as wellas

curve entire the denote to curve"" term

single the use shall we,simplicity For

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: (3) able curveDifferenti

)()()(

)()(

)()()(

tyi tx tz

bta

ty& tx

ti y tx tC: z

write weand , in continuous

arethey and exist if abledifferenti

be to said is curve The

smooth. or regular be

to said is (arc) curvea such then , If 0)( tz

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:(4) ve/arcSmooth curPiecewise

if smooth piecewise be to said is

,

curve The

bt, atyi tx t C:z )()()(

interval.-sub each on smooth is

that such of

intervals-sub of no. finite a exists there

Ca,b ,ba,....,,aa,a,a n ,][][][ 1211

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curve. the of point multiple called is point

a Such itself. touches or intersects

it whichat points havemay curveA

(5) :/ curvesimpleorcurvearcJordan

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i.e.,curve, simple a called is

POINTS MULTIPLE NO having curveA

itself, crosses nor itself touches

neither it if simple be to said iscurve a

.)()(

)()()(

2121 tttz tz

ti y tx tC:z

whenever

if simple be to said is

curve the i.e.,

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curve. Jordan a or curve closed simple

a be to said is then ,

that fact the for except simple is

curve the If

Cbzaz

bt, ati y tx tC:z

)()(

)()()(

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vetiable cura differenLength of (6).

(arc). curve bledifferenia a be

Let bt, atyi txtC: z )()()(

22)(

)()()(

tytxtz

tyi txtz'

and

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.

)(

C

dttz'Lb

a

curvetheoflengththecalledis

Then

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:(7).Contour

end. to end joined arcs smooth of

number finite of consistingarc an i.e.

arc, smooth piecewise a is ContourA

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.

)(

2 bzzazz

C

bt, atzz

point ato point

a from extending contour a denotes

Let

1

. on

continuous piecewise is i.e., on

continuous piecewise be Let

bta

tzfC

zf

))((

)(

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C

b

adttztzfdzzf

C f

)()(

:follows as

along of integral contour or

integral line the define weThen

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btatzC

dttztzfdzzfb

aC

),(:

,)())(()(1

:Properties

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C C

dzzfzdzzfz

z

)()(

2

00

0 then constant,ais If

C C

dzzgdzzf

dzzgzf

)()(

)()( )3(C

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atbtzzCzz

Czz

bta,tzC:z

),(:

)()4(

12

21

i.e. to from

extendedisthen,to from extended

is contour the If

O

z1

z2 C

x

y

- C dzzfdzzf

CC

And

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btctzzC

cta,tz:zC

bt; atzC:z

CCC

),(:&

)(

)(

,)5(

2

1

21 where Let

O X

Y

C1 C2

z2

C C C

dzzfdzzfdzzf

1 2

Then

z1

z3

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10,)(:

)(

Re)(

titttzC

dz,zf

z zf

C

where

evaluate then , Let

1Ex.

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2

1

1

)(.Re

)())(()(

1

0

1

0

i

dtit

dttztz

dttztzfzdzfC

b

a

(1,1) C

Y

X

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C

i

dzzf

ezC

z

zzf

evaluateThen

Let

Ex.2

.2,2:

&2

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idedz

ez

θi

θi

2

2:

Soln

= = 2

C

iez 2

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C

dzzfI )(

die.e

eπ

π

iθ

iθ

iθ

.22

222

2

12 dei i

i24

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34