Lecture 30
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Transcript of Lecture 30
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BITS Pilani Pilani Campus
MATH F112 (Mathematics-II)
Complex Analysis
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Lecture 30-34
Integrals
Dr Trilok Mathur,
Assistant Professor,
Department of Mathematics
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. 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