Chapter 3 Integral of Complex Function

§3.1 Definition and Properties

§3.2 Cauchy Integral Theorem

§3.3 Cauchy’s Integral Formula

§3.4 Analytic and Harmonic Functions

Review

1

( ) lim ( )nb

i ia ni

f x dx f x

1

( , ) lim ( , )n

i i in

iD

f x y d f

1

( , , ) lim ( , , )n

i i i ini

f x y z dV f V

1 1

( , ) ( , ) lim[ ( , ) ( , ) ]n n

i i i i i iL ni i

P x y dx Q x y dy P x Q y

n

iiiii

nSfdSzyxf

1

),,(lim),,(

1

( , ) lim ( , )n

i i iL ni

f x y ds f s

§3.1 Definition and Properties

0 1 2 1, , , , , , ,k k nz z z z z z

1 ( 1, 2, , )k k kz z k n

11 1

( )( ) ( ) , n n

n k k k k kk k

S f z z f z

1where k k kz z z

11

Let be the length of arc , max{ }k k k kk n

S z z S

( 0)

when & 0, lim exist.nn

n S

1. Def. C smooth (or piecewise smooth)

f : C→C.

( ) integrable on , the limit is called the integral

of ( ) along .

f z C

f z C

C1( 0)

Denote ( )d lim ( ) (3.1.1)n

k kn

k

f z z f z

C —— path of integral f —— integrandZ —— integration variable

the limit is called the integral of f along C,

denoted by .

If C is closed, we can write .

C( )df z z

C( )df z z

2. EvaluationLet i , ( ) ( , ) i ( , ), =1,2,k k k k k k k ky f u y v y k n

1 1 1i ( i )k k k k k k kz z z x y x y

1 1( ) i( ) ik k k k k kx x y y x y

1 1

( ) [ ( , ) i ( , )]( i )n n

k k k k k k k kk k

f z u v x y

1 1

[ ( , ) ( , ) ] i [ ( , ) ( , ) ]n n

k k k k k k k k k k k kk k

u x v y v x u y

1

d d lim [ ( , ) ( , ) ]n

k k k k k kC nk

u x v y u x v y

1

d d lim [ ( , ) ( , ) ]n

k k k k k kC nk

u x v y u x v y

TH 3.1.1 ( ) ( , ) i ( , ) integral on Cf z u x y v x y ( )d d d i d d

C C Cf z z u x v y v x u y (3.1.2

)Corollary: ( )is integral on C if ( ) is continuous on smooth or piecewise smooth arc C.

f z f z

( ) ( ) i ( ) [3.1.3], :z z t x t y t t

( )d ( ), ( ) '( ) ( ), ( ) '( ) dC

f z z u x t y t x t v x t y t y t t

i ( ), ( ) '( ) ( ), ( ) '( ) dv x t y t x t u x t y t y t t

( ), ( ) i ( ), ( ) ( '( ) i '( ))du x t y t v x t y t x t y t t

( ( ) '( )df z t z t t

(3.1.4)

Ex.1 010

d, C: ( 0), ,

( )nC

zz z r r n N

z z

C is counterwise clock.

010

2 , if 0d is independent of , .

0, if 0( )nC

i nzz r

nz z

center of a circle

radius

0

2

1 1100

2 2

0 0

parameter equation of is: ,0 2 ,i

i

n i nnc

inn in n

C z z re

dz ired

r ez z

i id e d

r e r

3. Properties

① ( )d ( )dC C

f z z f z z

② ( )d ( )d

C Ckf z z k f z z

③ ( ) ( ) d ( )d ( )dC C C

f z g z z f z z g z z ④

1 2 1 2

( )d ( )d ( )dC C C C

f z z f z z f z z

⑤ ( )d ( ) ds ML, ( ) M on C

C Cf z z f z f z

the length of C(Pf. P38)

⑥ ( )d independent of parametric representation of CC

f z z

①2

C : , [0,1]x t

ty t

1 20

2 11iI [2 i ] (2 i)d

3t t t

② C : , [0,2]2

x ttty

2 20

1 2 11iI [ i ] (1 i)d

2 2 3

tt t

2I dC

z zEx.

③ 1 2 1C=C C C : , [0, 2], 0 i0

x xx z x

y

2

2C : , [0,1], 2 i

xy z y

y y

1 20

2 11iI [2 i ] (2 i)d

3t t t

2 2 1 20 0

8 11i 2 11id (2 i) id 2

3 3 3x x y y

Note: I independent of integration path.

Ex.3.1.21 2

1Evaluate

1z

zI dz

z

On circle 1 2,

1 1 2 1( 1 2) 2.

1 2 2

the length of circle 1 2 is 4 ,

8 .

z

z zz

z

z L

I LM

Ex.3.1.32

2

Evaluate ( ) , :

(1). (0,1) (1,2) along 1;

(2). (0,1) (1,1) (1,2), .

Cz dz C

A B y x

A N B ANB

2

2 2

1 22 2

0

1 14 2 5 3

0 0

(1).The parametric equation of 1is:

( 1), : 0 1, ( 1), (1 2 ) ,

( ) ( 1) (1 2 )

(3 3 1) ( 2 4 4 )

3 10.

5 3

C

y x

z x i x x z x i x dz xi dx

z dz x i x xi dx

x x dx i x x x dx

i

1 2

12 2 2 2

0

2 2

1

2 .The parametric equation of is: , : 0 1,

the parametric equation of is: 1 , :1 2,

( ) ( ) ( ) ( 1 2 )

7 7[2 (1 ) ] .

3 3

C C C

AN z x i x

NB z iy y

z dz z dz z dz x xi dx

y y i dy i

Note: integration of f(z) dependent on integration path.

§3.2 Cauchy Integral Theorem

d d indepentent of integration pathC

P x Q yP Q

y x

—— continuous

( )d d d i d dC C C

f z z u x v y v x u y v u

y x

v u

x y

C-R equation

TH.3.2.1(Cauchy TH)

( )analytic on (simply connected domain)

( ) 0, .C

f z D

f z dz whereC D

TH 3.2.2 P42

1

1 2 0

( )d ( )d ( )dz

C C zf z z f z z f z z

upper limit

lower limit

0

( ) ( )d -primitive function antiderivativez

zF z f z z

( )analytic on (simply connected domain)

( ) along on is independent of integral path,

and it is only determined by starting point and end point.

then:

C

f z D

f z dz C D

TH 3.2.3 P45

0

( ) analytic on (simply connected domain)

( ) ( )d is analytic function in ,and '( ) ( ).z

z

f z u iv D

F z f z z D F z f z

0 0 0 0 0

0 0 0 0

( , ) ( , )

( , ) ( , )

( , ) ( , )

( , ) ( , )

Pf: ( ) ( )d

( , ) ( , ),

where ( , ) , ( , ) .

integration of ( )is independent of integration path,

so ar

z x y x y

z x y x y

x y x y

x y x y

F z f z z udx vdy i vdx udy

P x y iQ x y

P x y udx vdy Q x y vdx udy

f z

e ( , ) ( , ),

, , , , , .

( ) ( , ) ( , )is analytic function,

and '( ) ( ).

x y x y x y y x

x x

P x y and Q x y

P u P v Q v Q u P Q P Q

F z P x y iQ x y

F z P iQ u iv f z

Def.3.2.1

Properties

① G anti derivative of f on D G analytic on D

② G1and G2 anti derivative of f on D

G1=G2 +constant on D.

TH 3.2.5 (Fundamental Theorem of Contour Integral)

( ) is anti derivative of ( ) on , if '( ) ( ) on .G z f z D G z f z D

0 1

( )analytic on (simply connected domain),

'( ) ( ) on , , ,

f z D

G z f z D z z 0

1 0( )d ( ) ( )z

zf z z G z G z

Ex.3.2.15sin( 2 1)

23 1

.1

z z

z

eI dz

z

5sin( 2 1)

2

The singular points are outside 3 1,

so is analytic on and in 3 1,1

.3.2.1 0.

z z

z i z

ez

zTH I

Ex.3.2.22 2 2

1

1(1). ; (2). along the right curve.

i i

i iz dz dz

z

3

2 2 2 2 2 3 311

1 14(1). | [(2 2 ) (1 ) ] ( 1 ).

3 3 3

i iii

zz dz i i i

(2).Log function is multivalued,analytic on except , 0.

1( )

1| (ln arg 2 ) |

[arg( ) arg( )] .

k

i i ii ii

z x x

dLnz

dz z

dz Lnz z i z k iz

i i i i

C

Generalized Cauchy theorem in multi-connected domains

1 2 nT C C C C

TH 3.2.5 D multi connected with multi closed contours

Γ,f(z) analytic in D and on Γ.

1 2

( )d ( )d 0nC C C C

f z z f z z

1

or ( )d ( )dk

n

C Ck

f z z f z z

1

n=1, ( )d ( )dC C

f z z f z z -Deformation Theorem

2

1(1)

1C C C

dz dzdz

z z z z

2 11C C

dz dz

z z

2 2

0

i i

Closed Deformation Theorem

Ex.3.2.3

2

1 1 1( )

1z z z z

2

1, is positive oriented simple closed curve,

and 1 is in curve .

Cdz C

z zz C

Solutions:

C

1C2C

0 1

(2) .3.2.5TH

1 22 2 2

1C C C

dz dzdz

z z z z z z

1 1 2 21 1C C C C

dz dz dz dz

z z z z

0 2 2 0i i

0

Homework:

P59-60: A1-A7

00 0

( ) ( )Closed Deformation Theorem

C z z

f z f zdz dz

z z z z

0 0f z f z ��������������

0

0 00

1( ) d 2 π ( ).

z z

f z z if zz z

DC

0z

§3.3 Cauchy’s Integral Formula ＆ High Order Derivative

0 , and ( ) is analytic on D,then:z D f z

Analysis:

1. Cauchy’s Integral Formula: (TH 3.3.1)

0( ) analytic on C and on C, inside f z z C

00

1 ( )( ) d

2 i C

f zf z z

z z

Pf. ∵f(z) continuous at z0,

00, 0, z z

0

0

( ) ( ) ,

Let , :

f z f z

R k z z R C

0 0

( ) ( )d d

C K

f z f zz z

z z z z

0 0

0 0

( ) ( ) ( )d d ( 0)

K K

f z f z f zz z

z z z z

00

0 0

( ) ( )( ) ( )d d d 2

K K K

f z f zf z f zz s s

z z z z R

00

( )d 2 i ( )

C

f zz f z

z z

1 ( )

( ) d , : inside points of C2 i C

ff z z z

z

Note 1. f(z) on D depend on f(z) on C

D: domain

2. f =g analytic on C f =g on D

3. f: → C analytic.

00

1 ( )( ) d

2 i C

f zf z z

z z

2 i

00

1( e )dt

2tf z r

average of over [0,2 ]f

dzz

dzz zz

44 3

12

1

1

.62.21.2 iii

4

1 2

1 3zdz

z z

4

1 2,along the positive oriented

1 3

circle 4.

zdz

z z

z

Ex:

Ex: : ( 1, 2)

( 1)( 2)

z

C

edz C z r r

z z z

0 1,when r

C

z

dzz

zz

e

)2)(1(

izz

ei

z

z

0

)2)(1(2

Solution:

0

21 CC

2 1

)2(C

z

dzz

zz

e

i

3C

2

2C1

1C

1 2,when r

ie

i3

2

1)2(

2

z

z

zz

eii

3 2)1(

3

2C

z

dzzzz

e

ie

i

2)1(

23

2

z

z

zz

eii

ei

ie

ie

i33

2 2

2,when r

321 CCC

2. Existence of higher derivative

TH 3.3.2. f analytic on C & on D,

0 z D

0 10

! ( )( ) d , 1, 2,

2 i ( )n

nC

n f zf z z n

z z

Pf. n=1

0 00 0

1 ( ) 1 ( )( ) d , ( ) d

2 i 2 iC C

f z f zf z z f z z z

z z z z z

0 0

0 0

( ) ( ) 1 ( ) ( )d d

2 i C C

f z z f z f z f zz z

z z z z z z z

0 0

1 ( )d

2 i ( )( )C

f zz

z z z z z

2 20 0 0

1 ( ) 1 ( )d d

2 i ( ) 2 i ( ) ( )C C

f z zf zz z

z z z z z z z

I

3

ML0, ( 0)z z

d

20 0

1 ( )I d

2 ( ) ( )C

zf zz

z z z z z

2

0 0

( )1ds

2 C

z f z

z z z z z

0 20

1 ( )'( ) d

2 i ( )C

f zf z z

z z

Note. f(z) analytic on D

f (n)(z) exist on D & analytic on D. n=1,2,

…

-the difference with real function

5

cos(1). not analytic on 1in .

( 1)

but cos analytic in .

zz C

z

z C

.12

)(cos)!15(

2d

)1(

cos 5

1

)4(5 | i

zi

zz

zz

C

5 2 2

cos e(1) d ; ( 2) d ; | | 1,

( 1) ( 1)

C:positive oriented circle.

z

C C

zz z z r

z z

Ex:

2 2(2)

( 1)

z

C

edz

z

1 22 2 2 2( 1) ( 1)

z z

C C

e edz dz

z z

21

2

2

2

2

)(

)(

)(

)(C

z

C

z

dziz

iz

e

dziz

iz

e

2 22

( ) ( )

z z

z i z i

e ei

z i z i

)4

1sin(2 i

1 2C C C

2C

1C

§3.4 Analytic and Harmonic Function

continuous on D), and

Def. real

harmonic

on D, if

D is domain, : D , C R ( , )x y2( , ) C (D) ( , , , exist and x y xx yyx y

=0, xx yy Laplace Equation

is called harmonic function on D.

Def. u, v harmonic on D, v is harmonic conjugate of u

if ' ' , ' ' on D. (C-R equation)x y y xu v u v

( , )x y

Note. v harmonic conjugate of u u harmonic conjugate of v

i.e. u+iv analytic on D v+iu analytic on D

2 2 2

2 2Ex. ( ) i 2 analytic on 2 +i ( ) is not analytic on

f z z x y xyxy x y

CC

Properties:

(1). , , .?

u iv analytic on D u vharmonic on D

(2). i , analytic on D

  , harmonic on D & harmonic conjugate of .

f u v

u v v u

TH.3.4.2

(3). v harmonic conjugate of u -u harmonic conjugate of v

i.e. u+iv analytic on D v-iu analytic on D

(4). v harmonic conjugate of u on D u harmonic conjugate of v on D

u, v constants on D.

(5). v1,v2 harmonic conjugate of u on D v1=v2 + constant on D.

Pf. u+iv1 analytic on D, u+iv2 analytic on D i(v1-v2) analytic on D v1-v2 analytic on D (real)

v1-v2 =constant.

Question:

Does u have a harmonic conjugate (ux=vy , uy=-vx) on

D?

Does there exist an analytic f :D →C, u=Re f ?

(v=Im f )Ans. No in general .yes if D is simply connected.

D simply connected domain, u harmonic on D,

find , ( ) i analytic.v f z u v

d d d d dv v u u

v x y x yx y y x

0 0

( , )

( , )( , ) d d independent of integral path

x y

x y

u uv x y x y

y x

0 00( , ) '( , )d '( , )dy C

x y

y xx yv x y u x y x u x y

'( ) ' - ' '( ) ' - 'x y y xf z u iu or f z v iv integration of '( ).f z

Similarly,

0 00( , ) '( , )d '( , )dy C

x y

y xx yu x y v x y x v x y

Ex.3.4.2Prove 2 aharmonic function

and ( ) satisfying ( ) -1.

v xy

f z u iv f i

2 2

2 2

2 2

2 2

2

2

2 2

2 2 2 2

2 , 0; 2 , 0.

0 2 is a harmonic function.

2 2 ( ).

2 ' ( ) .

2 .

( ) 2 ( ) ,

( ) -1

v v v vy x

x x y y

v vv xy

x y

u vx u xdx x g y

x y

v uy g y g y y C

x y

u xdx x y C

f z x y C i xy x iy C z C

f i C

20. ( ) .f z z

Solution:

Ex.3.4.3 2 2is a harmonic function on Re( ) 0,

xu z

x y

1( ) satisfying (1 ) .

2

if z u iv f i

' '

1 0

2 2

2 2 2 2 2 2 20 0

2 2

(1). ( , ) ( ,0) ( , )

( ) .( )

1( ) .

1 1(1 ) 0 ( ) .

2

x y

y x

y y

v x y u x dx u x y dy C

y x y ydy C dy C C

x y y x y x y

x iyf z Ci iC

x y z

if i C f z

z

2 2 2 2

' '2 2 2 2 2 2 2 2 2 2

21

2 2 1(2). '( ) ,

( ) ( ) ( )

1 1( ) 1 .

1 1 1(1 ) ( ) .

2

x y

z

y x xyi y x xyif z u iu

x y x y x y z

f z dz iC iCz z

if i C f z

i z

method2

method1

Homework:

P60-61: A8-A17