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![Page 1: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/1.jpg)
1
(1) Indefinite Integration
(2) Cauchy’s Integral Formula
(3) Formulas for the derivatives of an analytic function
Section 5
SECTION 5Complex Integration II
![Page 2: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/2.jpg)
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
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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
![Page 4: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/4.jpg)
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
![Page 5: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/5.jpg)
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
![Page 6: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/6.jpg)
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
![Page 7: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/7.jpg)
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.......
![Page 8: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/8.jpg)
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)
![Page 9: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/9.jpg)
9
Section 5
Question:
dzz
z
2
22 1
sin
Can you evaluate the definite integral
![Page 10: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/10.jpg)
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
![Page 11: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/11.jpg)
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
![Page 12: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/12.jpg)
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
![Page 13: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/13.jpg)
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
![Page 14: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/14.jpg)
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
![Page 15: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/15.jpg)
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
![Page 16: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/16.jpg)
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
![Page 17: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/17.jpg)
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
![Page 18: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/18.jpg)
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
![Page 19: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/19.jpg)
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
![Page 20: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/20.jpg)
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
![Page 21: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/21.jpg)
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
![Page 22: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/22.jpg)
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*
![Page 23: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/23.jpg)
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)
![Page 24: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/24.jpg)
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
![Page 25: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/25.jpg)
25
Section 5
0zz
x
yz
![Page 26: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/26.jpg)
26
Section 5
0zz
x
yz
![Page 27: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/27.jpg)
27
Section 5
0zz
x
y
0z
z0zz
centre
![Page 28: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/28.jpg)
28
Section 5
0zz
x
y
0z
z0zz
centre
radius
![Page 29: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/29.jpg)
29
Section 5
231 jz
Question:
x
y
![Page 30: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/30.jpg)
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
![Page 31: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/31.jpg)
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
![Page 32: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/32.jpg)
32
Section 5
Cj
j
D
C z
dz
12
)(2)(
00
zfjdzzz
zf
C
jz 0let
CC jzjz
dz
z
dz
))((12
![Page 33: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/33.jpg)
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
![Page 34: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/34.jpg)
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
![Page 35: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/35.jpg)
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
![Page 36: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/36.jpg)
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.........
![Page 37: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/37.jpg)
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
![Page 38: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/38.jpg)
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
![Page 39: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/39.jpg)
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”
![Page 40: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/40.jpg)
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
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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
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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
![Page 43: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/43.jpg)
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
![Page 44: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/44.jpg)
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
![Page 45: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/45.jpg)
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
![Page 46: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/46.jpg)
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
![Page 47: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/47.jpg)
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
![Page 48: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/48.jpg)
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
![Page 49: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/49.jpg)
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)
![Page 50: 1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.](https://reader036.fdocuments.us/reader036/viewer/2022062518/56649eca5503460f94bd91f9/html5/thumbnails/50.jpg)
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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