ANTIDERIVATIVES Let f(z) be continuous function in a domain D. If there exists a function F(z) such that
then F(z) is called an
antiderivative of f(z) in D.
D,in z allfor )()( zfzF
Remark1: An antiderivative of a given function f is an analytic function. Remark 2: An antiderivative of a given function f is unique except for an additive complex constant.
Theorem: Suppose that a function f(z) is continuous on a domain D. If any one of the following statement is true, then so are the others:
i. f(z) has an antiderivative F(z) in D;
ii. the integrals of f(z) along
contours lying entirely in D and extending from any fixed point z1 to any fixed point z2 all have same value;
iii.the integral of f(z) around
closed contours lying entirely in D all have value zero.
Corollary:
D.in z allfor )()(
D.in f(z) of tiveantideriva
an is F(z) and D,domain
ain continuous is f(z)Let
zfzF
).()()(
Then D.in ENTIRELY
lying and ,z and z joining
contour any is C and D,in points
any two be z and zLet
12
21
21
zFzFdzzfC
Example:
.)( tiveantideriva
an has )( that Note
.
evaluate totiveantiderivaan Use2/
z
z
i
i
z
ezF
ezf
dze
i
ee
iFiFdze
ii
i
i
z
1
1
)()2/(
2/
2/
Cauchy - Goursat Theorem:
If a function f is analytic at all points
interior to and on a simple closed
contour C, then
0)( C
dzzf
Example: If C is any simple closed contour, in either direction, then
0)exp( 3 C
dzz
because the function )exp()( 3zzf is analytic
everywhere.
Defn: A simply connected
domain D is a domain
such that every simple
closed contour within it
encloses only points of D.
The set of points interior to a
simply closed contour is an
example.
A domain that is not
simply connected is said
to be multiply connected
for example, the annular
domain between two
concentric circles.
The Cauchy – Goursat
theorem for a simply
connected domain D is
as follows:
Theorem: If a function f is analytic throughout a simply connected domain D, then
0)(
C
dzzf
for every closed contour C lying in D.
Result: Let C1 and C2 denote
positively oriented simple
closed contours, where C2 is
interior to C1 .
If a function f is analytic in
the closed region consisting
of those contours and all
points between them, then
21
)()(CC
dzzfdzzf
Ex.1 Evaluate
dzzf
C
when zzezf , C: |z|=1.
Ans: 0 (Why??)
Ex.2 Evaluate
dzzf
C
when
.2:,4
sin2
zCz
zzzf
Ans: 0 (Why??)
Qs 3/154. Let C0 denote the circle Rzz 0 , taken counter clockwise
using the parametric representation
izz Re0 for C0 to derive the following integrations:
no. realany is 0 where
,sin2
)(
,...2,1,0)( (b)
2 )(
0
10
0
10
0 0
a
aa
iRdzzzc
ndzzz
izz
dza
a
C
a
C
n
C
Sol. We have Rzz 0
iddz
zzi
i
.Re
Re0
a)
ii
id
zz
dzI
i
i
C
2
Re
.Re
0 0
b)
tion)simplifica(after 0
Re11
1
00
dieR
dzzzI
inin
C
n
c)
dieR
dzzzI
iaia
C
a
Re11
1
00
aSina
Ri a2
Exercise:
• Does Cauchy – Goursat Theorem hold separately for the real or imaginary part of an analytic function f(z) ? Justify your answer.
Cauchy Integral Formula
Czz
dzzf
izf .
)(
2
1)(
then sence,
positive in the taken C,contour
closed simple aon and inside
everywhere analytic be fLet
00
Derivative Formula
then C,
ointerior tpoint any is If sence.
positive in the taken C,contour closed
simple aon and inside everywhere
analytic is ffunction a that Suppose
0z
C
C
zz
dzzf
izfb
zz
dzzf
izfa
,)(
)(
2
)!2()()
,)(
)(
2
1)()
30
0
20
0
Cn
n
zz
dzzf
i
nzf
c
.)(
)(
2
)!()(
)
10
0)(
Theorem:
If f(z) is analytic at z0, then its
derivatives of all orders exist at
z0 and are themselves analytic
at z0.
Qs.1(a)/163: Let C denote the positively oriented boundary of the square whose sides lie along the
lines 2x and 2y . Evaluate
the following integral .
)8(
cos2
Czz
dzz
Ans : i/4.
Qs. 2(b)/163: Find the value of the integral of g(z) around
the circle 2 iz in the positive sense when
22 4
1
zzg
.
16
2
12
224:
22
2222
iz
CC
izdz
di
iziz
dz
z
dzSol
Qs.4/163: Let C be any simple closed contour, described the positive sense in the z- plane and write
dz
wz
zzwg
C
3
3 2
Show that
iwwg 6
when w is inside C and that
0wg
when w is outside C.
wfi
dzwz
zfwg
zzzf
C
2
2
,
Then .2)(Let
C. inside is Let w :I Case
3
3
zzzf 23
wwf
zzf
zzzf
6
6
3 2
iwwgI 6
Case 2. When w is outside C,
then by Cauchy Goursat
Theorem 0wg .
Qs. 5/163: Show that if f is analytic within and on a simple closed contour C and z0 is not on C, then
dzzz
zfdz
zz
zf
CC
200
Sol. Let
dzzz
zfI
dzzz
zfI
C
C
20
2
01 and
Case I: Let z0 is inside C, then
0
01
2
20
zfi
zfidzzz
zfI
zzC
and
0
20
2
2 zfi
dzzz
zfI
C
21 II .
Case II: Let z0 is outside C
Then I1 = I2 = 0.
(WHY ???)
Morera’s Theorem:
D.in analytic is f(z) then D,in
lying Ccontour closedevery for
,0f(z)dz
if and Ddomain ain throughout
continuous is f(z)function a If
C
LIOUVILLE’S THEOREM If f is entire and bounded in the complex plane, then f(z) is constant throughout the plane.
Fundamental Theorem of Algebra
.0)P(z
such that zpoint on least at exist
thereisThat zero. oneleast at
has )1( degree of )0(
...)(
polynomialAny
0
0
2210
nna
zazazaazP
n
nn
Theorem: Suppose that
(i) C is a simple closed contour, described in the counter-clockwise direction,
(ii) Ck (k = 1, 2, …., n) are finite no. of simple closed contours, all described in the clockwise direction, which are interior to C and whose interiors are disjoint.
.0f(z)dzf(z)dz
then,C ointerior t points for the
except Con and within points
all of consistingregion closed
t the throughouanalytic is f(z) If
1
n
k kCC
k
Ex. Evaluate C zz
dz
)1( 2 for all
possible choices of the contour C that does not pass through any of the points 0,
i .
Solution: Case 1. Let C does not enclose 0,
i . Then
.012 TheoremCGby
z
dzI
C
Case 2a. Let C encloses only 0. Then
ifi
zzf
z
dzzfzz
dzI
C
C
2)0(2
)1(
1)(,
0
)()1(
2
2
Exercise: Case 2b. Let C encloses only i. Ans: I = -i Case 2c. Let C encloses only -i. Ans: I = -i
Case 3 a). Let C encloses only 0, -i. then
iCCizizz
dz
izizz
dzI
))(())((0
where C0 and C-i are sufficiently small circles around 0 and –i resp.
iC
C
dzizizz
dzz
iziz
)()(
1
))((1
0
iii
ii
i
)2(
1)2(
1)2( 2
Case 3 b). Let C encloses only 0, i. then
iCCizizz
dz
izizz
dzI
))(())((0
where C0 and Ci are sufficiently small circles around 0 and i resp.
i
iiii
dzizizz
dzz
izizI
iC
C
2.
122
)()(
1
))((1
0
Case 3 c). Let C encloses only -i, +i. Then
i
iii
iii
dzizizz
dzizizz
I
i
i
C
C
22.
12
2.
12
)()(
1
)(1
Case 3 d). Let C encloses all of the points 0, -i, +i. Then
0
2
)()(
1
)(1
1
1
0
2
iii
dzizizz
dzizizz
dzz
zI
iC
iCC
0
2
)()(
1
)(1
1
1
0
2
iii
dzizizz
dzizizz
dzz
zI
iC
iCC