Chapter 7. Applications of Residues Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University Email ： [email protected] Office ： # A313 [email protected]

Chapter 7. Applications of Residues

Weiqi Luo (骆伟祺 )School of Software

Sun Yat-Sen UniversityEmail ： [email protected] Office ： # A313

mailto:[email protected]

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Evaluation of Improper Integrals Improper Integrals From Fourier Analysis Jordan’s Lemma Definite Integrals Involving Sines and Cosines

2

Chapter 7: Applications of Residues

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Improper Integral

78. Evaluation of Improper Integrals

3

0 0( ) lim ( )

R

Rf x dx f x dx

2

11 2

0

0( ) lim ( ) lim ( )

R

RR Rf x dx f x dx f x dx

If f is continuous for the semi-infinite interval 0≤x<∞ or all x, its improper integrals are defined as

when the limit/limits on the right exists, the improper integral is said to converge to that limit/their sum.

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Cauchy Principal Value (P.V.)


4

. . ( ) lim ( )R

RRPV f x dx f x dx

2

11 2

0

0( ) lim ( ) lim ( )

R

RR Rf x dx f x dx f x dx

0

0. . ( ) lim ( ) lim[ ( ) ( ) ]

R R

R RR RPV f x dx f x dx f x dx f x dx

0

0lim ( ) lim ( )

R

RR Rf x dx f x dx

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Example Observe that

However,


5

2

. . lim lim[ ] | 02

R RRRR R

xPV xdx xdx

2

11 2

0

0lim lim

R

RR Rxdx xdx xdx

2

11 2

2 20

0lim[ ] | lim [ ] |2 2

RR

R R

x x

1 2

2 21 2lim lim2 2

R R

R R

An Odd Function

No limits

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Suppose f(x) is an even function

and assume that the Cauchy principal value exists, then


6

( ) ( ), ( )f x f x x

0

1 1( ) lim[ ( ) ] [ . . ( ) ]

2 2

R

RRf x dx f x dx PV f x dx

( ) . . ( )f x dx PV f x dx

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Evaluation Improper Integrals of Ration Functions

where p(x) and q(x) are polynomials with real coefficients and no factors in common.

assume that q(z) has no real zeros but at least one zero above the real axis, labeled z1, z2, …, zn, where n is less than or equal to the degree of q(z)


7

( ) ( ) / ( )f x p x q x

( ) ( ) / ( )f x p x q x ( ) ( ) / ( )f z p z q z

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8

1

( ) ( ) 2 Re ( )k

R

nR

R z zkC

f x dx f z dz i s f z

lim ( ) 0R

RC

f z dz

When

1

. . ( ) 2 Re ( )k

n

z zk

PV f x dx i s f z

01

( ) Re ( )k

n

z zk

f x dx i s f z

When f(x) is even

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Properties

79. Example

9

0 1

0 1

...( )( )

( ) ...

nn

mm

a a z a zp zf z

q z b b z b z

Let

where m≥n+2, an≠0, bm≠0, then we get lim ( ) 0R

RC

f z dz

1 2

1 2 01 2

1 2 0

| ... |( ) | || ( ) | | |

( ) | | | ... |

nnn n n

m mm m m

a a z a z a zp z zf z

q z z b b z b z b z

1 2

1 2 01 2

1 2 0

| | | ... || |

| | | | | ... |

nnn n n

m mm m m

a a z a z a zz

z b b z b z b z

<|an|, R-> ∞

<1/2|bm|, R-> ∞| |4

| | | |n

m nm

a

z b

1

| | | |4 1| ( ) | | | ( ) 4

| | | | | | | |R

n nm n m n

m mC

a af z dz R

R b R b 0

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Example

79. Example

10

2

60 1

xdx

x

6 1z Firstly, find the roots of the function

2exp[ ( )], ( 0,1, 2,...5)

6 6k

kC i k

None of them lies on the real axis, and the firstthree roots lie in the upper half plane

And 6-2=4≥2 2

6lim 0

1R

RC

zdz

z

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Example(Cont’)

79. Example

11

2 22

6 60

lim 2 ( Re ), 11 1k

R

RR z Ck

x zdx i s R

x z

Here the points ck are simple poles of f, according to the Theorem 2 in pp. 253, we get that

2

6

1 1 12 ( )

1 6 6 6 3

xdx i

x i i i

2

60 1 6

xdx

x

Even Function

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pp. 267

Ex. 3, Ex. 4, Ex. 7, Ex. 8

79. Homework

12

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Improper Integrals of the Following Forms

80. Improper Integrals From Fourier Analysis

13

( )sinf x axdx

( ) cosf x axdx

OR

where a denotes a positive constant

( ) ( ) / ( )f x p x q x

where p(x) and q(x) are polynomials with real coefficients and no factors in common. Also, q(x) has no zeros on the real axis and at least one zero above it.

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Improper Integrals In Sec. 78 & 79,


14

( )sinf x axdx

( ) cosf x axdx

( )sinf z azdz

( ) cosf z azdz

The moduli increase as y tends to infinity

( ) ( ) cos ( )sinR R R

iax

R R R

f x e dx f x axdx i f x axdx

( )| | | | | || | 1 iaz ia x iy iax ay aye e e e eThis moduli is bounded in theupper plane y>0 (a>0), and islarger than 0.

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Example Let us show that

Because the integrand is even, it is sufficient to show that the Cauchy principal value of the integral exists and to find that value. We introduce the function

The product f(z)ei3z is analytic everywhere on and above the real axis except at the point z=i.


15

2 2 3

cos3 2

( 1)

xdx

x e

2 2

1( )

( 1)f z

z

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Example (Cont’)


16

33

12 22 ( ) , ( 1)

( 1)R

i xR i z

RC

edx iB f z e dz R

x

31 Re [ ( ) ]i z

z iB s f z e

33

2 2

( )( ) , ( )

( ) ( )

i zi z z e

f z e zz i z i

the point z = i is evidently a pole of order m = 2 of f (z)ei3z, and

1 3

1'( )B i

ie

32 2 3

cos3 2Re ( )

( 1)R

R i z

RC

xdx f z e dz

x e

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Example (Cont’)


17

32 2 3

cos3 2Re ( )

( 1)R

R i z

RC

xdx f z e dz

x e

2 2 2 2

1 1| ( ) | | | ,

( 1) ( 1)R Rf z M Mz R

3 3| Re ( ) | | ( ) |R R

i z i zR

C C

f z e dz f z e dz M R 4 3

2 22

4 2

1

1 1( 1) (1 )

R R RR

R R

2 2 3

cos3 2

( 1)

xdx

x e

0, when R ∞

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Theorem Suppose that

a) a function f (z) is analytic at all points in the upper half plane y ≥ 0 that are exterior to a circle |z| = R0;

b) CR denotes a semicircle z = Reiθ (0 ≤ θ ≤ π), where R > R0;

c) for all points z on CR, there is a positive constant MR such that

81. Jordan’s Lemma

18

| ( ) | , lim 0

R RRf z M M

Then, for every positive constant a,

lim ( ) 0R

iaz

RC

f z e dz

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19

sin

0

, ( 0)Re d RR

The Jordan’s Inequality

Consider the following two functions siny 2y

2sin ,0

2

sin 2 /0 R Re e

/2 /2sin 2 /

0 0

R Re d e d (1 )

2Re

R

/2 sin

0 2Re d

R

sin

0

Re dR

sinΘ is symmetric with Θ=π/2

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20

0

( ) (Re )exp( Re )( Re )

R

iaz i i i

C

f z e dz f ia i d

sin| (Re ) | ,| exp( Re ) | ,| Re | i i aR iRf M ia e i R

According to the Jordan’s Inequation, it follows that

sin

0

| ( ) |R

iaz aR RR

C

Mf z e dz M R e d

a

The final limit in the theorem is now evident since MR0 as R∞

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Example Let us find the Cauchy principal value of the integral

we write

where z1=-1+i. The point z1, which lies above the x axis, is a simple pole of the function f(z)eiz, with residue


21

2

sin

2 2

x xdx

x x

21 1

( )2 2 ( )( )

z zf z

z z z z z z

11

1

1 1

izz eB

z z

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Example (Cont’)

which means that


22

122 ( ) , ( 2)

2 2R

R ixiz

R C

xe dxiB f z e dz R

x x

12

sinIm(2 ) Im( ( ) )

2 2R

Riz

R C

x xdxiB f z e dz

x x

| Im( ( ) ) | | ( ) |R R

iz iz

C C

f z e dz f z e dz 2

1 1

| ( ) | | |( )( ) ( 2)

R

z Rf z M

z z z z R

| | 1, ( 0)iz ye e y

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Example (Cont’)


23

2

| Im( ( ) ) | | ( ) |2

(1 )R R

iz izR

C C

f z e dz f z e dz M R

R

0

However, based on the Theorem, we obtain that

12

sin. . Im(2 ) (sin1 cos1)

2 2

x xdxPV iB

x x e

lim ( ) 0R

iz

RC

f z e dz

2lim lim 0

( 2)RR R

RM

R

Since

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pp. 275-276

Ex. 2, Ex. 4, Ex. 9, Ex. 10


24

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Evaluation of the Integrals

85. Definite Integrals Involving Sines and Cosines

25

2

0(sin ,cos )F d

The fact that θ varies from 0 to 2π leads us to consider θ as an argument of a point z on a positively oriented circle C centered at the origin.

Taking the radius to be unity C, we use the parametric representation

, (0 2 )iz e idzie iz

d

sin2

i ie e

i

cos2

i ie e

1

sin2

z z

i

1

cos2

z z

2

0(sin ,cos )F d

1 1

( , )2 2C

z z z z dzF

i iz

dzd

iz

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Example Let us show that


26

2

20

2, ( 1 1)

1 sin 1

da

a a

2

20

2 /

1 sin (2 / ) 1C

d adz

a z i a z

where C is the positively oriented circle |z|=1.

2 2

1 2

1 1 1 1( ) , ( )

a az i z i

a a

1 2| | 1,| | 1, (| | 1)z z a

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Example (Cont’)


27

2

20

2 /

1 sin (2 / ) 1C

d adz

a z i a z

1 2

2 /

( )( )C

adz

z z z z

Hence there are no singular points on C, and the only one interior to it is the point z1.

The corresponding residue B1 is found by writing

1 2 1 2

2 / ( ) 2( ) , ( ( ) )

( )( )

a z af z z

z z z z z z z z

This shows that z1 is a simple pole and that

1 1 21 2

2 1( )

1

aB z

z z i a

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Example (Cont’)


28

1 2

2 /

( )( )C

adz

z z z z

2

20

2 /

1 sin (2 / ) 1C

d adz

a z i a z

1 2

22

1iB

a

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pp. 290-291

Ex. 1, Ex. 3, Ex. 6


29

Documents

Chapter 7. Applications of Residues Weiqi Luo ( 骆伟祺 ) School of Software Sun Yat-Sen University Email ： [email protected] Office ： # A313 [email protected]