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(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(3
1
)3/8(1)21(
)84()443()21(
)21()22(
)()(
1
0
22
1
0
2
1
0
j
jj
dtttjttj
dtjtjt
dtdt
dztzfdzzf
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
3
8)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
3
2
3
11)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
z
C
z
z
C
z
z
C
z
z
C
z
z
C
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 )(zfdz
dF
0z
1z
)219(3
1
332
3
21
321
2
2 jzz
dzzzjz
j
e.g.
Not only that, but.......
8
Section 5Examples
(1)
j
jj
zdzzj
j
j
j
097.23
sinh2)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
211
2
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 5
Question:
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)(
00
zfjdzzz
zf
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)(
00
zfjdzzz
zf
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
z
2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(
00
zfjdzzz
zf
C
The Cauchy Integral formula
422
2
jdzz
z
C
14
Section 5Example
C
dzz
z
2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(
00
zfjdzzz
zf
C
The Cauchy Integral formula
422
2
jdzz
z
C
15
Section 5Example
C
dzz
z
2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(
00
zfjdzzz
zf
C
The Cauchy Integral formula
422
2
jdzz
z
C
16
Section 5Example
C
dzz
z
2
2
Evaluate the integral where C is
20 z
Singular point inside !
becomes
or jdzz
z
C
82
2
21 z
)(2)(
00
zfjdzzz
zf
C
The Cauchy Integral formula
422
2
jdzz
z
C
17
Section 5Illustration of Cauchy’s Integral Formula
)(2)(
00
zfjdzzz
zf
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
z
z
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)(
00
zfjdzzz
zf
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)(
00
zfjdzzz
zf
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)(
00
zfjdzzz
zf
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
jdzzC
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)(
00
zfjdzzz
zf
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
jdzzC
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
dtdt
dztzfdzzf jtjt
C
21
)()(2
0
2
0
2
0*
23
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
28
Section 5
0zz
x
y
0z
z0zz
centre
radius
29
Section 5
231 jz
Question:
x
y
30
Section 5
)(2)(
00
zfjdzzz
zf
C
Examples
Evaluate the following integrals:
C jz
dz(1) where C is the circle z 2
jz 0let
1)( zflet
f (z) is analytic in D and C encloses z0
1)( 0 zf
C j
D
jjz
dz
C
2
31
Section 5
)(2)(
00
zfjdzzz
zf
C
C z
dz
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 jzjz
dz
z
dz
))((12
32
Section 5
Cj
j
D
C z
dz
12
)(2)(
00
zfjdzzz
zf
C
jz 0let
CC jzjz
dz
z
dz
))((12
33
Section 5
Cj
j
D
C z
dz
12
)(2)(
00
zfjdzzz
zf
C
jz 0letjz
zf
1
)(let
CC jzjz
dz
z
dz
))((12
34
Section 5
Cj
j
D
C z
dz
12
)(2)(
00
zfjdzzz
zf
C
jz 0letjz
zf
1
)(let2/)( 0 jzf
CC jzjz
dz
z
dz
))((12
35
Section 5
C z
dz
14(3) where C is the circle zj1C
j
j
1 1Here we have
CC jzjzzz
dz
z
dz
))()(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
zf
z
dz )(
14 ))(1)(1(
1)(
jzzzzf
where
4)2)(1)(1(
1)()( 0
j
jjjjfzf
Now
2)(2
)(
1 00
4
zfjdzzz
zf
z
dz
CC
36
Section 5
Question:
Evaluate the integral
C
z
jz
dze
1where 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 z
zdz
1
tan2(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(
111
12
zz
zBzA
z
B
z
A
z
2/1,2/11
0)(
BABA
zBA
38
Section 5
CCC
dzz
zdz
z
zdz
z
z
1
tan
2
1
1
tan
2
1
1
tan2
C
11 2/
1tan)(
tan)(
1
0
0
zf
zzf
z
)1tan()(
tan)(
1
0
0
zf
zzf
z
jjdzz
z
C
785.9)1tan()1tan(2
12
1
tan2
39
Section 5
0
!
2)(1
0 z
n
n
Cn dz
fd
n
jdz
zz
zf
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
zfjdzzz
zf
Generalisation of Cauchy’s Integral Formula
C zzC dz
zdjdz
z
z
dz
zzdjdz
z
zz
2
2
2
3
1
2
2
2
00
cos
2
cos,
32
1
3
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
zfjdzzz
zf
C
Example
Evaluate the integral
C
z
dzz
e2
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
jz
dze
C
z2
22
41
Section 5
)(2)(
)(02
0
zfjdzzz
zf
C
Example
Evaluate the integral
C
z
dzz
e2
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
jz
dze
C
z2
22
42
Section 5
)(2)(
)(02
0
zfjdzzz
zf
C
Example
Evaluate the integral
C
z
dzz
e2
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
jz
dze
C
z2
22
43
Section 5
)(2
2
)(
)(03
0
zfj
dzzz
zf
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
zf
zf
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
dzzz
zf )(with f (z) analytic inside and on C, except at zzo - equals
(2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzz
zf )(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
dzzz
zf )(with f (z) analytic inside and on C, except at zzo - equals
(2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzz
zf )(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
dzzz
zf )(with f (z) analytic inside and on C, except at zzo - equals
(2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzz
zf )(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
dzzz
zf )(with f (z) analytic inside and on C, except at zzo - equals
(2)
( Cauchy’s Integral Theorem )
( Cauchy’s Integral Formula )
Ck
o
dzzz
zf )(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)(
00
zfjdzzz
zf
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 )(zfdz
dF
(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
r
Mnzf
!)( 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
