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1
(1) Indefinite Integration
(2) Cauchy’s Integral Formula
(3) Formulas for the derivatives of an analytic function
Section 5
SECTION 5Complex Integration II
2
value of the integralbetween two pointsdepends on the path
1C
dzz
y
x
j1
0
jdzzC
1no real meaning to
j
dzz1
0
Section 5
3
integrate the function along the path Cjoining 2 to 12j as shown
2)( zzf
Example
1022)( ttjttz
)219(31
)3/8(1)21(
)84()443()21(
)21()22(
)()(
1
0
22
1
0
2
1
0
j
jj
dtttjttj
dtjtjt
dtdtdztzfdzzf
C
y
x
j21
0
C
2
Section 5
4
integrate the function along the path CC1 C2 joining 2 to 12j as shown
2)( zzf
Example
202)( tttz
38)44(
)1()2()(
2
0
2
2
0
2
dttt
dttdzzfC
y
x
j21
0 21C
2C
Along C1:
0
2
2dxx
alongreal axis !
102)( ttjttzAlong C2:
jdttj
dtjtjtdzzfC
32
311)211(
)21()2()(
1
0
2
1
0
2
Section 5
5
3/)219(2 jdzzC
y
x
j21
0
3/)219(2 jdzzC
value of the integralalong both paths is
the same
2
coincidence ??
Section 5
6
Dependence of Path
1z
2z1C
2C0)()(
21
CC
dzzfdzzf
Suppose f (z) is analytic ina simply connected domain D
D
by the Cauchy Integral Theorem
2
2
1
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
alongalong
alongalong
alongalong
)()(
)()(
0)()(
C
z
zC
z
z
C
z
zC
z
z
C
z
zC
z
z
dzzfdzzf
dzzfdzzf
dzzfdzzf
1z
2z
1C2C
note:if they intersect,we just do thisto each “loop”,one at a time
Section 5
7
Section 5Integration (independence of path)
Consider the integral dzzfz
z1
0
)(
If f (z) is analytic in a simply connected domain D, and z0
and z1 are in D, then the integral is independent of path in D
)()()( 01
1
0
zFzFdzzfz
z
where )(zfdzdF
0z
1z
)219(31
33 2
3
21
321
2
2 jzzdzzzjz
j
e.g.
Not only that, but.......
8
Section 5Examples
(1)
jjj
zdzz j
j
j
j
097.23sinh2)sin(2
sincos
the wholecomplex plane
C
(2) ?1
0
dzzj
( f (z) not analytic anywhere - dependent on path )
(3) jz
dzz
j
j
j
j
2112
f (z) analytic in
this domain
(both 1z2 and 1z are not analytic at z0 - the path of integration C must bypass this point)
9
Section 5Question:
dzz
z
2
22 1
sin
Can you evaluate the definite integral
10
Section 5More Integration around Closed Contours ...We can use Cauchy’s Integral Theorem to integrate aroundclosed contours functions which are
(a) analytic, or (b) analytic in certain regions
For example,
0C z
dzC
f (z) is analytic everywhereexcept at z0
But what if the contour surrounds a singular point ?C
?C z
dz
11
)(2)(0
0
zfjdzzzzf
C
Section 5Cauchy’s Integral Formula
Let f (z) be analytic in a simply connected domain D. Then forany point z0 in D and any closed contour C in D that encloses z0
D
0z
C
12
Section 5Cauchy’s Integral Formula
)(2)(0
0
zfjdzzzzf
C
Let f (z) be analytic in a simply connected domain D. Then forany point z0 in D and any closed contour C in D that encloses z0
D
0z
C
13
Section 5Example
C
dzz
z2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(0
0
zfjdzzzzf
C
The Cauchy Integral formula
422
2
jdzzz
C
14
Section 5Example
C
dzz
z2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(0
0
zfjdzzzzf
C
The Cauchy Integral formula
422
2
jdzzz
C
15
Section 5Example
C
dzz
z2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(0
0
zfjdzzzzf
C
The Cauchy Integral formula
422
2
jdzzz
C
16
Section 5Example
C
dzz
z2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(0
0
zfjdzzzzf
C
The Cauchy Integral formula
422
2
jdzzz
C
17
Section 5Illustration of Cauchy’s Integral Formula
)(2)(0
0
zfjdzzzzf
C
Let us illustrate Cauchy’s Integral formulafor the case of f (z)z and z0 1
10 z
C DSo the Cauchy Integral formula
becomes
121
jdz
zz
C
1)1( f
or jdzz
z
C
21
f (z) is analytic everywhere,so C can be any contour in thecomplex plane surrounding thepoint z1
18
Section 5Another Example
)(2)(0
0
zfjdzzzzf
C
jz 0
C D
The Cauchy Integral formula
becomes
j
C
z
ejdzjz
e 2
or j
C
z
jedzjz
e 2
C
z
dzjz
eEvaluate where C is any closed contour
surrounding zj
f (z) is analytic everywhere
19
Section 5Another Example
)(2)(0
0
zfjdzzzzf
C
jz 0
C D
The Cauchy Integral formula
becomes
j
C
z
ejdzjz
e 2
or j
C
z
jedzjz
e 2
C
z
dzjz
eEvaluate where C is any closed contour
surrounding zj
f (z) is analytic everywhere
20
Section 5Another Example
)(2)(0
0
zfjdzzzzf
C
Let us illustrate Cauchy’s Integral formulafor the case of f (z)1 and z0 0
00 z
C DSo the Cauchy Integral formula
becomes
121 jdz
zC
or jdzzC
21
f (z) is a constant, and so is entire,so C can be any contour in thecomplex plane surrounding theorigin z0
21
Section 5Another Example
)(2)(0
0
zfjdzzzzf
C
Let us illustrate Cauchy’s Integral formulafor the case of f (z)1 and z0 0
00 z
C DSo the Cauchy Integral formula
becomes
121 jdz
zC
or jdzzC
21
f (z) is a constant, and so is entire,so C can be any contour in thecomplex plane surrounding theorigin z0
22
Section 5
Cut out the point z0 from the simply connected domain by introducinga small circle of radius r - this creates a doubly connected domain inwhich 1z is everywhere analytic.
From the Cauchy Integral Theorem as appliedto Doubly Connected Domains, we have
jdzzC
21
note: see section 4, slide 6
*
11
CC
dzz
dzz
C
*C
Let us now prove Cauchy’s Integral formulafor this same case: f (z)1 and z0 0
But the second integral, around C*, is given by
jdtjdtrjeer
dtdtdztzfdzzf jtjt
C
21)()(2
0
2
0
2
0*
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What does the equation mean ?
Section 5Equations involving the modulus
1z
1
122
22
yx
yxz
equation of a circle22
02
0 )()( ryyxx
x
y
zz
mathematically:
(these are used so that we can describe paths(circles) of integration more concisely)
24
Section 5
x
y
Example
12 z
1)2(
)2(
2)(2
22
yx
yjx
jyxz
1)2( 22 yx
equation of a circle22
02
0 )()( ryyxx
z
25
Section 5
0zz
x
yz
26
Section 5
0zz
x
yz
27
Section 5
0zz
x
y
0z
z0zz
centre
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Section 5
0zz
x
y
0z
z0zz
centre
radius
29
Section 5
231 jz
Question:
x
y
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Section 5
)(2)(0
0
zfjdzzzzf
C
Examples
Evaluate the following integrals:
C jzdz
(1) where C is the circle z 2
jz 0let1)( zflet
f (z) is analytic in D and C encloses z0
1)( 0 zf
C j
D
jjz
dz
C
2
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Section 5
)(2)(0
0
zfjdzzzzf
C
C zdz
12(2) where C is the circle zj1
We need a term in the form 1(z z0) so we rewrite the integral as:
First of all, note that 1(z21) hassingular points at zj.
The path C encloses one of these points, zj.We make this our point z0 in the formula
Cj
j
D
CC jzjzdz
zdz
))((12
32
Section 5
Cj
j
D
C z
dz12
)(2)(0
0
zfjdzzzzf
C
jz 0let
CC jzjzdz
zdz
))((12
33
Section 5
Cj
j
D
C z
dz12
)(2)(0
0
zfjdzzzzf
C
jz 0letjz
zf
1)(let
CC jzjzdz
zdz
))((12
34
Section 5
Cj
j
D
C z
dz12
)(2)(0
0
zfjdzzzzf
C
jz 0letjz
zf
1)(let
2/)( 0 jzf
CC jzjzdz
zdz
))((12
35
Section 5
C zdz
14(3) where C is the circle zj1C
j
j
1 1Here we have
CC jzjzzzdz
zdz
))()(1)(1(14
The path C encloses one of the four singular points, zj.We make this our point z0 in the formula
CC
dzjz
zfz
dz )(14 ))(1)(1(
1)(jzzz
zf
where
4)2)(1)(1(1)()( 0
jjjj
jfzf
Now
2)(2)(
1 00
4
zfjdzzzzf
zdz
CC
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Section 5Question:
Evaluate the integral
C
z
jzdze1
where C is the circle z 2
(i) Where is C ?
(ii) where are the singular point(s) ?
(ii) what’s z0 and what’s f (z) ? Is f (z) analytic on and inside C ?
(iii) Use the Cauchy Integral Formula.........
37
Section 5
C zzdz
1tan
2(4) where C is the circle z3/2
tanz is not analytic at /2, 3/2, , but thesepoints all lie outside the contour of integration
The path C encloses two singular points, z1.To be able to use Cauchy’s Integral Formula we mustonly have one singular point z0 inside C.
C
112/3 2/
Use Partial Fractions:
)1)(1()1()1(
1111
2
zz
zBzAzB
zA
z
2/1,2/11
0)(
BABA
zBA
38
Section 5
CCC
dzz
zdzz
zdzz
z1
tan21
1tan
21
1tan
2
C
11 2/
1tan)(tan)(
1
0
0
zfzzf
z
)1tan()(tan)(
1
0
0
zfzzf
z
jjdzz
z
C
785.9)1tan()1tan(212
1tan
2
39
Section 5
0
!2)(
10 z
n
n
Cn dz
fdn
jdzzzzf
For example,
More complicated functions, having powers of zz0, can betreated using the following formula:
Note: when n0 we haveCauchy’s Integral Formula: 0
)(2)(
0z
C
zfjdzzzzf
Generalisation of Cauchy’s Integral Formula
C zzC dzzdjdz
zz
dzzzdjdz
zzz
22
2
31
2
2
2
00
cos2
cos,3213
f analytic on andinside C, z0 inside C
This formula is also called the “formula for the derivatives of an analytic function”
40
Section 5
)(2)(
)(02
0
zfjdzzzzf
C
Example
Evaluate the integral
C
z
dzze
2
where C is the circle z 2
C
00 zlet
zezf )(let
f (z) is analytic in D, and C encloses z0
0
0 )(
)(
ezf
ezf z
D
jzdze
C
z2
2 2
41
Section 5
)(2)(
)(02
0
zfjdzzzzf
C
Example
Evaluate the integral
C
z
dzze
2
where C is the circle z 2
C
00 zlet
zezf )(let
f (z) is analytic in D, and C encloses z0
0
0 )(
)(
ezf
ezf z
D
jzdze
C
z2
2 2
42
Section 5
)(2)(
)(02
0
zfjdzzzzf
C
Example
Evaluate the integral
C
z
dzze
2
where C is the circle z 2
C
00 zlet
zezf )(let
f (z) is analytic in D, and C encloses z0
0
0 )(
)(
ezf
ezf z
D
jzdze
C
z2
2 2
43
Section 5
)(2
2)(
)(03
0
zfjdzzzzf
C
Another Example
Evaluate the integral
C
dzjz
z3
2
where C is the circle z 2
C
jz 0let
2)( zzf let
f (z) is analytic in D, and C encloses z0
2)(2)(
0
zfzf
D
jjz
dzz
C
2)( 3
2
44
Section 5Summary of what we can Integrate
C
dzzf )( with f (z) analytic inside and on C - equals 0(1)
C o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo - equals (2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo
(3)
( The Formula for Derivatives )
)(2 0zfj
45
Section 5Summary of what we can Integrate
C
dzzf )( with f (z) analytic inside and on C - equals 0(1)
C o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo - equals (2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo
(3)
( The Formula for Derivatives )
)(2 0zfj
46
Section 5Summary of what we can Integrate
C
dzzf )( with f (z) analytic inside and on C - equals 0(1)
C o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo - equals (2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo
(3)
( The Formula for Derivatives )
)(2 0zfj
47
Section 5Summary of what we can Integrate
C
dzzf )( with f (z) analytic inside and on C - equals 0(1)
C o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo - equals (2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzzzf )( with f (z) analytic inside and on C,
except at zzo
(3)
( The Formula for Derivatives )
)(2 0zfj
48
Section 5
What can’t we Integrate ?
(singularities at 2 inside C)
C
zdzze / where C is the unit circle
(singularity at 0 inside C)
e.g.
Functions we can’t put in the form of our formulas:
1z
C
zdztan where C is e.g. 2z
49
Section 5Topics not Covered
(2) Proof of Cauchy’s Integral Formula
(3) Proof that the derivatives of all orders of an analytic function exist - and the derivation of the formulas for these derivatives
)(2)(0
0
zfjdzzzzf
C
(use the MLinequality in the proof)
(use the MLinequality in the proof)
(1) Proof that the antiderivative of an analytic function exists
)()()( 01
1
0
zFzFdzzfz
z
where )(zfdzdF
(use Cauchy’s Integral Formula and the MLinequality in the proof)
50
Section 5(4) Morera’s Theorem (a “converse” of Cauchy’s Integral Theorem)
(5) Cauchy’s Inequality
“If f (z) is continuous in a simply connected domain D and if 0)( C
dzzf
for every closed path in D, then f (z) is analytic in D”
nn
rMnzf !)( 0
)( 0zr
CMzf on)(
C
(proved using the formula for the derivatives of an analytic function and the MLinequality)
(6) Liouville’s Theorem“If an entire function f (z) is bounded in absolute value for all z, then f (z) must be a constant” - proved using Cauchy’s inequality