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Ch7.2 Calculus of Residues講者： 許永昌 老師
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ContentsResidue TheoremEvaluation of Definite IntegralsCauchy Principle values
Some poles on the integral path.Pole expansion of Meromorphic FunctionProduct Expansion of Entire function
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Residue Theorem( 請預讀 P378~P379)
Laurent expansion
Closed contour
integration
Cauchy’s integral theorem
Residue Theorem
0
n
nn
f z a z z
1
1
2f z dz a
i
10
2for analytic function
except for isolated poles.
f z dzi
STOP TO THINK: How about multivalent function?
1,
1Each around ,
2 ii
i i zCC z f z dz a
i
1,
1
2 izf z dz a
i
Besides, {residue at z} -
S{residues in the finite z-plane}
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ResidueResidue:
In some books, e.g. J. Bak and D.J. Newman, Complex Analysis, it denotes by Res(f ; zi).
Exercise: Hint: Homework 7.1.1
a-n.
-a-n. (Q: 為何不是 -a-1*0=0? 所以要搞清楚 Laurent expansion.) 用 ordinary series expansion 比較快找到值。
1
Res ;- n
az a z
1
Res ;0- nz a z
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Evaluation of Definite Integrals ( 請預讀 P379~P384)
4 types we will discussed here:
Hint: (1) |z|=1, (2) cosq=(z+z-1)/2, (3)sinq=(z-z-1)/(2i)
Related to Jordan’s Lemma.
2
1 0sin ,cos ,I f d
2 ,
(1) no pole on the path (2) lim 0 for upper (lower) half planez
I f z dz
zf z
3 ,
(1) no pole on the path (2) 0 (3) lim 0 for upper half plane
iax
z
I f z e dz
a f z
4 0
keyhole contour
, Use a multivalent function ln to solve it.
ln , ln 0 when 0 & .2
I f z dz z
iz f z dz z z f z z z
0lim ln 0.zz z
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Exercises ( 請預讀 P379~P384)
Step 1: find the singular points.Step 2: find a suitable contour.Step 3:
For branch point, we must consider the branch cut. For poles, find a-1 on each pole.
Step 4: Residue theorem.
2
1 0
2 2
4 30
, 11 cos
,1
,1
dI
dxI
xdx
Ix
Code: quadgk(@(z)(1./(1+z.^2)),-inf,inf)quadgk(@(z)(1./(1+z.^3)),0,inf)
小心， (1) 確定 q 的範圍 (2) 一整圈為 2p 。
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Upper half circle whose radius is
iazR
R
I f z e dz
Jordan’s Lemma( 請預讀 P383)
If (1) a>0, aR, (2) lim|z| f (z)=0, 0 arg(z) p,We get limR|IR|=0,
Proof:
cos sin
0
sin
,max 0
22
,max 0
,max
2
1
.
i iaR aR iR
aR
C
aR
C
aR
C
I f R e e e iR e d
R f e d
R f e d
f ea
a
0 pi/2 pi0
0.5
1
1.5
2
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Cauchy Principle Value( 請預讀 P384)
Situation:Some singular points are directly on the
contour of integration.We define
When f (x0) is finite, the principle value limit is unnecessary.
For a simple pole, Therefore,
0
00lim
x
xf x dx f x dx P f x dx
semicircle
1
2 circlef z dz f z dz
.C C Semicircle
P f x dx f z dz f z dz f z dz
C
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Pole Expansion of Meromorphic Functions ( 請預讀 P390~P391)
Mittag-Leffler theorem: A meromorphic function can be written as
If all the poles of this function are simple poles, we get
,
1
1make the sum converge entire functionpk
k nn
n k
k kk
a
z z
f z S z z E z
STOP TO THINK: 請問與 Taylor and Laurent expansion 有何不同 ?
1
1
1 10 if and ,
where is a radius circle includes ,..., but no other poles.
kk kC
k k k
k k k
f z f b f z R k Nz z z
C R z z
1
11
1
00 ' 0 ...
!
if and .k
pp pk
pk k n
pkC
z f b zf z f zf
p z z z
f z R k N
We need to prove that their remainder converges to zero.
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Pole Expansion of Meromorphic Functions (continue)Proof for |f(z)|<eRk case:
The remainder for |f(z)|<eRkp+1 case:
1
1
if is analytic at 0 and z 1. Laurent expansion at ,2. Cauchy's integral formula,3. Simple pole at
1
2
Res ;0 Res ; Res ;
0 1 1 1
k
n
k C
F
k
nn
k
nn n n
f z
fI d
i z
F F z F z
f f zb
z z z z z z
.
.
nz
max on
1 1 12
2 2kkk
k
k kCR zk Cf R
f fI d R
z R z
1
1.
2 kk pC
fI d
i z
記得，是 F(w) 的 residue 。
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ExercisesTest the pole expansion for:
Test them with the remainder to understand the meaning of |f(z)|<eRk
p+1.
11 ,
1f z
z
11 ,
1f z z
z
1
1.
2 kk pC
fI d
i z
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Example ( 請預讀 P391)
Pole expansion of cotangent: p cotpz=
Method I: Its pole is located at z=n, nZ. We will find that Choose Rk=k+0.5, we get (I did not test it) |f(z)|
<eRk. Therefore, based on Mittag-Leffler theorem, we get
Method II: The product expansion of
cos ln sin.
sin
z d z
z dz
1
coslim 1, they are simple poles whose 1 for all .
sinz n
zz n a n
z
0
1 1 1cos
nn
zz z n n
2
2 21
sin 1 .n
zz z
n
13
1
,kn
q
kk
f z g z z z
Product Expansion of Entire Functions ( 請預讀 P392~P394)
An entire function with zeros at z1,…, zn can be written as where g(z) is an entire function with no zero.Questions:
How to find the number of zeros in a region? How to do this product expansion for an entire
function?Key concept:
1
entire functionSimple poles
ln ' '.
nk
k k
d f z f z g zq
dz f z z z g z
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Product Expansion of Entire Functions (continue)
How to find the number of zero points?
How to do the expansion? From the pole expansion, if |F(z)/Rk|<e, we get
1
entire functionSimple poles
ln ' '.
nk
k k
d f z f z g zq
dz f z z z g z
1
'1
2
1arg . arg ument
2
n
kCk
C
f zdz q
i f z
f z f z
代表繞一圈後 的 的變化。
1
' ' 0 1 1.
0 kk k k
f z fq
f z f z z z
01
' 0
0
1
' ' ' 0' ln ln 1
' 0 0
0 1 .k k
k
zk
kk k k
qf q zz
f z
k k
f z f z f q zzdz z q
f z f f z z
zf z f e e
z
小心：此處要求 '
.kg z
Rg z
'
ln ln argf z
dz f z f z i f zf z
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Example
Zeros: mp, m0. f(0)=1 f ’(0)=0
STOP TO THINK: cos(z) ?
sin:z
z
' 0
0
1
0 1 .k k
k
qf q zz
f z
k k
zf z f e e
z
,
0
' 1 1 1cot . 1.k
nn
f zz q
f z z z n n
2
, 10
sin1 1
z
n
n nn
z z zf z e
z n n
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Rouché’s theoremIf f(z) and g(z) are analytic inside and on a
closed contour C, and |g(z)|<|f(z)| on C, then f(z) and f(z)+g(z) have the same number of zeros inside C.Proof:
DC arg(1+g(z)/f(z))=0 DC arg f(z)=DC arg [f(z)+g(z)]Z-plane w-plane
1g z
wf z
1
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Homework1, 2, 3, 4, 6, 9, 14, 16, 21, 22
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NounsResidue of f(z) at z=z0: a-1, the coefficient
of (z-z0)-1 in the Laurent expansion: Res(f ; z0).
Jordan’s Lemma: P383Cauchy Principle Value: P384Pole Expansion: P390Product Expansion: P392
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