tch-prob 1

Chap 6 Residues and Poles

Cauchy-Goursat Theorem: c f dz 0 if f analytic.

What if f is not analytic at finite number of points interior to C Residues.

53. Residues

z0 is called a singular point of a function f if f fails to be analytic at z0 but is analytic at some point in every neighborhood of z0.

A singular point z0 is said to be isolated if, in addition, there is a deleted neighborhood of z0 throughout which f is analytic.

00 z z ε

tch-prob 2

Ex1.2 2

1 has isolated singnlar points 01z

z z , i(z )

Ex2. The origin is a singular point of Log z, but is not isolated

Ex3.

1sin( )

1singular points 0 and 1 2 . z

z z n , , ...n

not isolated isolated

When z0 is an isolated singular point of a function f, there is a R2 such that f is analytic in 0 2

0 z z R

tch-prob 3

Consequently, f(z) is represented by a Laurent series

1 20 2

0 0 0

0 2

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

0

( )1where 2 (

nn n

bb bnf z a z z z z z z z znz z R

f z dzbn i z

0

( 1, 2, ... )c 1)n

nz

and C is positively oriented simple closed contour

0 0 2around and lying in 0z z z R

When n=1, 12 ( ) (2)πi b f z dzc

The complex number b1, which is the coefficient of in expansion (1) , is called the residue of f at the isolated singular point z0.

0

1z z

0

Re ( )z z

s f z

A powerful tool for evaluating certain integrals.

tch-prob 4

湊出 z-2 在分母

1

)

116

1 1 1but4 4 2 ( 2)( 2) ( 2)

1 14 22( 2) 1 (

2( 1) 4( 2) 0 2 2

1201 2

16 2( 2)

.zz z- z

.zz-

n- n- z - z -nn

dz b ic z z-

8π i

2Re

1 has singular points at 0 24( 2)

it has Laurent series representation in 0 2 2

1based on (2) 24 4( 2) ( 2)z

s

z , zz z

z-

dz πi c z z z z

2 14( 2)

dz C : z-c z z Ex4.

0 2

tch-prob 5

1

)

2 3 1

1! 2! 3!1

1 1 1 1 12 1 02 2! 4 3! 61!

01 exp ( 02

analytic on and within

z

z

z z zz e ......

z e ..... z z z

b

dz c z

f C f

The reverse is not necessarily true.

0dz c

2

2

( )1show exp 0

where 1

1 is analytic everywhere except at the origin

dzc z

C : z

z

Ex5.

tch-prob 6

54. Residue Theorems

Thm1. Let C be a positively oriented simple closed contour. If f is analytic inside and on C except for a finite number of (isolated) singular points zk inside C, then

Cauchy’s residue theorem Res

Res

( ) 2 ( )1

pf:

( ) ( ) 01

but ( ) 2 ( )

z zk

k

z zk k

nf z dz πi f zc

k

n f z dz f z dz c c

k f z dz π i f zc

Z1

Z2

Z3

C

tch-prob 7

Ex1.

1 1 0

)

Re

5 2Evaluate where 21

Two singularities 0 1a. When 0 1

5 2 5 2 1 2 2(5 ( 1 )1( 1)

( ) 2z

s

z- dz C : zc z(z- )z , z

zz- z - - - z- z .....z z- zz z-

b B f z

1 2

)

b. when 0 1 1

5( 1) 35 2 1 . 1( 1) 1 ( 1)

3 2(5 [1 ( 1) ( 1) ..]1

b 3

5 2 2 (2 3) 10( 1)

z-

zz- zz z- z

- z - z- .....z

B

z dz π i π ic z z

tch-prob 8

C

C0

R1 R0

Thm2:

If a function f is analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour C, then

Pf:

1

0

10

1

2

is not the residue of at = ( )

1

)

0

From Laurent Theorem

( ) ( ) (3)

( )1where 2 1

( ) 2

Replace by in (3)

11 (

z

z

c f

nf z c z R z nnf z c dzn cπi nz

f z dz πi cc

z ,

f nz

z

22

1

1 20

)

Re )]

1(0

1 1[ (

n nn n

zs

c c zz z Rn

c fz z

1

2

)

now is the residue1 1of ( at 0

c

f zzz

20

1 1Re )]( ) 2 [ (z

szz

f z dz πi fc

tch-prob 9

Ex2.

0

)

Re )

5 2( )( 1)

1 1 5 2 5 2 1(2 1(1 )

5 2( 2)(1

5 3 3 (0 1)

1 1( 52

( ) 10z

s

zf zz z

z zf .z z zz zz z z ....)z z ...... zz f zz f z dz

πic

tch-prob 10

55. Three Types of Isolated Singular points

If f has an isolated singular point z0, then f(z) can be represented by a Laurent series

1 20 2

0 0 0

0 2

( ) ( ) ..... ....( ) ( )0

in a punctured disk 0

nn

bb bnf z a z zn z z z z z zn z z R

1 22

0 0 0

0.

The portion ..... ....( ) ( )

is called the principal part of at

nn

bb bz z z z z z

f z

tch-prob 11

(i) Type 1.

1 2

10

0 0

0 2

0

0 and 0

( ) ( ) ..............( ) ( )n 0

0

The isolated singular point

m m m

mm

b b b .......bbnf z a z zn z z z z

z z R

z

is called a pole of order .m

pole simple1 , m

Ex1.

0 1

2 ( 2)2 2 3 3 3 (0 2 )

2 2 2

Simple pole 1 at 2 3.

zz z z z -z z z

m z , b

tch-prob 12

Ex2.

3 5

4

3

3

0 1

(sinh 14 3! 5!

1 1 1 03! 5! 7!

1has pole of order 3 at 06

zz z z ....)zz

z z ........... zzzm z , b

(ii) Type 2

bn=0, n=1, 2, 3,……

20 0 1 0 2 0

0 2

( ) ) )( ) ( (0

0

z nf z a z a a z z a z z ...... nn

z z R

0z is known as a removable singular point.

* Residue at a removable singular point is always zero.

tch-prob 13

* If we redefine f at z0 so that f(z0)=a0

define

Above expansion becomes valid throughout the entire disk

0 2z z R

* Since a power series always represents an analytic function Interior to its circle of convergence (sec. 49), f is analytic at z0 when it is assigned the value a0 there. The singularity at z0 is therefore removed.

Ex3. [2 4 61 cos 1( ) 1 (1 )]

2 2 2! 4! 6!2 41 (0 )

2! 4! 6!

z z z zf z ....z z

z z ....... z

0

.

1when the value (0) is assigned, become entire,2

the point 0 is a removable singular point.sin* another example ( )

f f

z z f z z

tch-prob 14

(iii) Type 3:

Infinite number of bn is nonzero.

0z is said to be an essential singular point of f.

In each neighborhood of an essential singular point, a function assumes every finite value, with one possible exception, an infinite number of times. ~ Picard’s theorem.

tch-prob 15

has an essential singular point at

where the residue

00z

11b

* Note that exp 1 when (2 1) ( 0 1 2 )

1 1exp ( ) 1 when(2 1) (2 1)

z - z n π i n , , , ...

i z - z n πi n π

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

an infinite number of these points clearly lie in any given neighborhood of the origin.

)1* Since exp( 0 for any value of , zero is the exceptional

value in Picard's theorem.

zz

Ex4.

2)1 1 1 1 1 1 1exp( 1 0

! 1! 2!0 ...... znz zn z zn

tch-prob 16

* exp when (2 1/2) ( 0 1 2 )

1 1exp ( ) when(2 1/2) (2 1/2)

z i z n π i n , , , ...

i i z - z n πi n π

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

an infinite number of these points clearly lie in any given neighborhood of the origin.

* exp 1 when 2 ( 0 1 2 )

1 1exp ( ) 1 when2 2

( 0 1 2

z z nπ i n , , , ...

i z - z nπi nπ n , , , ..

).

tch-prob 17

56. Residues at Poles

identify poles and find its corresponding residues.

Thm. An isolated singular point z0 of a function f is a pole of order m iff f(z) can be written as

0)

( )( )(

zf z mz z

00

0

0

0

Res )

( )Res

where ( ) is and Moreover, ( ) ( if 1

( 1)and

analytic nonzero a

( ) if 2( 1

t

)!

z z

z z

z . f z z m

m- z f z m

m

z

tch-prob 18

0

0

( ) .( - )

, it has a Taylor series representation

Suppose ( )

Since ( ) is analytic at

m

z

z zf z

z z

20 00 0 0

( )( 1)010

0 00

0 01 2

00 0

0

'( ) ''( )( ) ( ) ( ) ( )1! 2!

( )( ) ( ) ( ) !( 1)!

'( ) /1! ''( ) / 2!( )( )

( )

( ) ( )

nmnm

n m

mm m

z zz z z z z z

zz z z z

z

z

z z znm

z zf zz zz z z

.......

.......

( )0

0 00

0 0(

(

0

1)

1)

0

0

( )( ) 0

!( )) 0, is a pole of order of ( ) and

( )

( )

Res ( ) .( 1)!

/( 1)!

Since (

m nn m

n m

m

z z

zz z z z

nz zz m f z

zf z

z m

m

z

Pf: “<=“

tch-prob 19

0 is a pole of order of , or ( ) has a Laurent series representationIf m f f zz

“=>”

1 20 2

0 0 0

0 2

( 0)( ) ( ) .....( ) ( )0

in a punctured disk 0

mmm

bb b bnf z a z zn z z z z z zn

z z R

0 0

0

The function defined by

( ) ( ) when ( )

when

has the power series representation

m

m

z z f z z zz

b z z

2 101 0 2 0 1 0

0) ) ) ( )( ) ( ( ( m nm m

nm mn

b a z zz b b z z b z z z z

0 2

0

0

.

Consequently, ( ) is analytic in that disk (sec.49)and, in particular at .

Also ( ) 0.

throughout

m

zz

z b

z z R

tch-prob 20

Ex1. 1( ) has an isolated singular point at 3 2 9zf z z i

z

( ) 1( ) where ( )3 3z zf z z

z i z i

3Re

( ) is analytic at 33 1 (3 0 a simple pole

63 6

another simple pole 33 residue6

z is

z z i i i)

ii

z - ii

tch-prob 21

Ex3.

0

0)

sinh( ) 4

To find residue at 0,( )can not write ( ) ( ) sinh4

since ( 0

zf zz

zzf z , z z

z z

Need to write out the Laurent series for f(z) as in Ex 2. Sec. 55.

0

sinh 1 1 1 14 3 3! 5!

10 is a pole of the third order, its residue 6

z

z

z ..........zz z

tch-prob 22

Ex4.

Since ( 1) is entire and its zeros are 2 ( 0 1 2 )

0 is an isolated singular point of 1( )

( 1)2 3

11! 2! 3!

2( 1) 1

z z e z n i n , , , ..... z

f z zz e

z z zze .... z

zzz e z (

2

)2! 3!

( ) 1Thus ( ) ( )2 2

12! 3!

z ..... z

z f z zz z z .......

1

Since ( ) is analytic at 0 and (0) 1 00 is a pole of the second order

1 2( ) 12! 3!(0)2 22(1 ) 02! 3!

z z , z

z ..... b ' -

z z ..... z

tch-prob 23

57. Zeros and Poles of order m

Consider a function f that is analytic at a point z0.

(From Sec. 40). 0( )( ) ( 1 2 ) exist atznf n , , .... z

0

0

0

0

)

)

( )

( )

( 0'( 0

( 1) 0( ) 0

If f z , f z :

m- f zm f z

Then f is said to have a zero of order m at z0.

0

0

)

.

Lemma: ( ) ( ( )

analytic and non-zero at

m f z z z g z

z

tch-prob 24

Ex1.

0

(

( 1) / when 0 is analytic at 0.

1 when 0

( ) ( 1)22 1 )

2! 3! has a zero of order 2 at 0

( )ze z z

zz

zf z z e

z z z ......

m z

g z

Thm. Functions p and q are analytic at z0, and 0( ) 0.p z

If q has a zero of order m at z0, then

( )( )

p zq z

has a pole of order m there.

0

0

)

)

( ) ( ( )

analytic and non zero

( ) ( ) ( )( ) (

mq z z z g z

p z p z /g zmq z z z

tch-prob 25

Ex2.0

1( ) has a pole of order 2 at 0( 1)z

f z zz e

Corollary: Let two functions p and q be analytic at a point z0.

0 0 0) ) ) If ( 0 ( 0 and ( 0 p z , q z , q' z

0

0

0 0

)Re

)

then is a simple pole of and

(( ) ( ) (z z

s

p(z)zq(z)

p zp z q z q' z

Pf:0 0

0

( ) ( ) ( ), ( ) is analytic ard non zero at ( ) ( )/ ( )( )

q z z z g z g z zp z p z g z

z zq z

Form Theorem in sec 56,0

0 0

0

0

)Res

)

))

((((

z z

p zp(z) q(z) g z

p z

q' z

0 0) )But ( ( g z q' z

tch-prob 26

Ex3. cos( ) cotsin

( ) cos sin both entire

zf z z z

p z z, q(z) z

The singularities of f(z) occur at zeros of q, or

( 0 1 2 )z n n , , , ...

Since ( ) ( 1) 0 ( ) 0 and ( ) 1 0n n p nπ , q nπ , q' nπ (- )

each singular point of is a simple pole,( ) ( 1)with residue 1( ) ( 1)

z nπ fnp nπBn nq' nπ

try tan z

tch-prob 27

Ex4

0 :

:

( )4 4

42 1

zf zz

iπz e i

tch-prob 28

58. Conditions under which

Lemma : If f(z)=0 at each point z of a domain or arc

0)( zf

containing a point z0, then in any neighborhood N0 of z0 throughout which f is analytic. That is, f(z)=0 at each point z in N0.

0)( zf

Pf: Under the stated condition, For some neighborhood N of z0

f(z)=0

Otherwise from (Ex13, sec. 57)

There would be a deleted neighborhood of z0 throughout which 0)( zf

0

inconsistent with ( ) 0 in a domain or arc containing .

f zz

arcZ0

N

N0

tch-prob 29

0)( zfSince in N, an in the Taylor series for f(z) about z0

must be zero.

Thus in neighborhood N0 since that Taylor series also represents f(z) in N0.

( ) 0f z

Z0Z若有一點 0)( zf

0則全不為

Ex13, sec 57

圖解Z

0Z全為 0

arc or domain 0 ,若在 為 則

Theorem. If a function f is analytic throughout a domain D and f(z)=0 at each point z of a domain or arc contained in D, then in D.0)( zf

0Z 1Z 2Z 3Z nZ

P

tch-prob 30

Corollary: A function that is analytic in a domain D is uniquely determined over D by its values over a domain, or along an arc, contained in D.

arc

D

domain( ) ( ) analytic in D( ) ( ) in some domain or arc contained in ( ) ( ) ( ) 0 in a domain or acr( ) 0 in D( ) ( ) in D

f z , g zf z g z Dh z f z -g z h zf z g z

Example: 2 2Since sin cos 1x x

along real x-axis (an arc)

2 2

2 2

( ) sin cos 1is zero along the real axis

( ) 0 throughout the complex plane sin cos 1 for all

z z

z z

f z

f zz

tch-prob 31

59. Behavior of f near Removable and EssentialSingular Points

Observation :

A function f is always analytic and bounded in some deleted neighborhood of a removable singularity z0.

00 z z ε

0 2 0

0 2

)

.

For is analytic in a disk when ( is

properly defined at such a point; and is then continuous in any closed disk when

Consequently, is bounded in that disk.(From sec. 14,)

f z - z R f z

fz- z ε ε R

f

0 must be bounded in 0 .f z - z ε

tch-prob 32

Thm 1: Suppose that a function f is analytic and bounded in some deleted neighborhood of a point z0. If f is not analytic at z0, then it has a removable singularity there.

00 z- z

Pf: Assume f is not analytic at z0.

The point z0 is an isolated singularity of f and f(z) is represented by a Laurent series

0 00

( ))

( ) throughout 0(0 1

n nn n

bf z a z- z z - z ε

z- zn n

If C denotes a positively oriented circle

0z- z

where

tch-prob 33

0

0

0

)

( )1 ( 1 2 )2 1(

since ( ) 0

1 22 1

since can be chosen arbitrarily small, 0 is a remorable singularity of .

f z dzb n , , ...n cπ i nz- z

f z M z - z

M nb . πρ Mρn π -nρ bn

z f

0ZC

Thm2.

Suppose that z0 is an essential singularity of a function f, and let w0 be any complex number. Then, for any positive number , the inequality

0( )f z w

is satisfied at some point z in each deleted neighborhood

0 00 of z z z

(a function assumes values arbitrarily close to any given number)(3)

tch-prob 34

Pf: Since z0 is an isolated singularity of f. There is a 0

0 z z

throughout which f is analytic.

Suppose (3) is not satisfied for any z there. Thus

00( ) when 0 -f z w z z

0

1( )( )

g zf z w

is bounded and analytic in 0

0 z z

According to Thm 1, z0 is a removable singularity of g. We let g be defined at z0 so that it is analytic there,

0 0 0) 1If ( 0 ( ) 0

( ) g z , f z w z z δ

g z

becomes analytic at z0 if it is defined there as

0 00

1( )( )

f z wg z

But this means that z0 is a removable singularity of f, not an essential one, and we have a contradiction.

tch-prob 35

0

0 0

)If ( 0 ( ) must have a zero of some finite order (sec. 57)at because is not identically equal to zero in .

g z , g z mz g z z δ

0 0, has a pole of order at .

So, again, we have a contradiction. Theorem 2 is proved.

1In view of ( )( )

f m zf z wg z

00( ( ) when 0 - )f z w z z