1 chap 6 residues and poles cauchy-goursat theorem: if f analytic. what if f is not analytic at...
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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
Singular points
C
殘值
除了那點 Z0 之外的小圈圈 ( 半徑為 ) 之內 f 都是可解析的
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 0
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.
0 20 z z R
R.O.C.
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
4
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. 0
6
More on Cauchy Integral Formula (1)Simply-Connected and Multiply-Connected
Simply ConnectedMultiply Connected
( ) ( ) 01 k
nf z dz f z dzc c
k
C1 C2
C( ) 0f z dzc
C
7
C0
C
........ same. dzcZ
dθπ θiei ρ dθ π iθρe c Z dz
π ic zdz
π iπiθρ e
iθiρρ c z
dzce
iθρe Z
π ic zdz to show
02
020
22200
2
0Zat except everywhere analytic is Z1 and
2200
sin
,0C circlea stmct con
2
More on Cauchy Integral Formula (2)Simply-Connected and Multiply-Connected
8
( )
0
( )0
0
0( )
00
( )
( ))
( ))
0) ( ) (!0
or (!0
Replace by -
( ) (!0
nn
nn
nn
z
z
g g z z z Rnn
f f z z z
nnz z z ,
f f z z z
nn
More on Cauchy Integral Formula (3)Simply-Connected and Multiply-Connected
f(z)cf(s) ds
π i ZS 2
1
)π i f(zc Z-Z
) dzf(zc Z-Z
f(z) dz0
20
0
0
π ic Z-Z
zz dc Z-Z
dzc Z
dz 20
)0
(
0
Then connection to Taylor Series…….
1
100 00( )1
00( )1
0 0
( )
(0)2
!
(0)( )
!
( ) ( )( )
( ) ( )
( )1( )2
wher
Nn N
c n Nn
nNn N
Nn
nNn
Nn
f s ds
s
fi
n
fz z
n
f s ds f s ds z zc cs z s z s
f s ds z z c s z s
f s dsf z ρc s zπi
0( ) ( )e
2 ( )N
N Nz f s dsz ρ cπi s z s
)π i f(zc Z-Z
) dzf(z
02
0
0
9
Why (chap4)
π
πρ ρ dz cZZ
)f(zf(z)
dzc ZZ
)f(zf(z)c dz
ZZ
)f(zf(z)btu
dzc ZZ
)f(zf(z) ) π i f(z - c dz
ZZf(z)
)π i f(z dzπZZ
)f(zc Z-Z dz)but f(z
dzc ZZ
)f(zf(z)
2
20 0
0
0 0
00 0
0
)5( 0 0
00
20
022
00
00 00
0 0
0
1
)π i f(zc Z-Z
) dzf(zc Z-Z
f(z) dz0
20
0
0
10
More on Cauchy Integral Formula (4)Simply-Connected and Multiply-Connected
( ) 0f z dzc
( ) ( ) 01 k
nf z dz f z dzc c
k
C1 C2C
C
C
)()( 10 CC
C1
...)()()()()( 43210
CCCCC
C
C1C2
)()()( 210 CCC
11
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
12
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
13
分解大突破
3,2
25)1(
1)1(
25
BA
zBzzA
z
B
z
A
zz
z
325
21
25
1)1(
25
1
0
z
z
z
zB
z
zA
z
B
z
A
zz
z
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
展開法 係數比較法 因式分解法
14
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
20
1 1Re )]( ) 2 [ (z
szz
f z dz πi fc
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
展開法 係數比較法 因式分解法
分別找 k 個 singular points 的 c-1(Residue)
1 個 z=0 的 residue g(z)
2
Z-1 Z10
15
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
16
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
17
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
18
(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
19
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.
20
* 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
21
(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.
22
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
23
* 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 , , , ..
).
24
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
25
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: “<=“
26
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
27
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
28
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
29
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
30
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
31
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
32
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
34
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
0
f(z)
連續周圍 2D, 3D
35
f(z0)
0z0
f(z0)
0z0
n
n
nn zzaw
00 )(
f(z0)
0z0
z
0)( 00 zfw
0
00 )(
ww
zzawn
n
nn
0)( 00 zfw
0)(0
0
n
n
nn zzaw
f(z0)
0z0
z
0
36
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
37
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
38
Sect. 59
• Reference, study for strengthen your theory background
• Is not covered in final exam
43
Cauchy Integral 補充
1. Cauchy-Goursat Theorem: c dzzf 0)( if f analytic.
12
2. )0
(2
0
)( zfic dzzz
zf
3
zz
X z0
f(z)
zz
zf
0
)(
!
)0
()(
210
)(
n
zn
fic dznzz
zf 3.
44
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
4
45
C
z
dzz
e23 for C: 1z
C
z
dzz
e4
2
)3(
3 for C: 2z
C
z
dzzz
e2
2
)1)(5(
3 for C: 2z
C
z
dzz
e2
2
)3(
1 for C: 1z
C
z
dzz
e2
21 for C: 13 z
1z
Ans:2πi·3
2z 2z
Ans:0