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Residue Theory: Evaluating Real Integrals using Contour Integration, Slides of Mathematics

The application of residue theory in evaluating real integrals through contour integration. The theory is based on the Laurent series expansion of analytic functions and the residue theorem. the calculation of residues, trigonometric integrals, and improper integrals involving trigonometric functions.

What you will learn

  • How to compute the residue of a function with a pole of order m?
  • What is the role of Laurent series expansion in residue theory?
  • How does the residue theorem help in evaluating real integrals?

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Ch.6: Residue Theory
Chapter 6: Residue Theory
Li, Yongzhao
State Key Laboratory of Integrated Services Networks, Xidian University
June 7, 2009
Ch.6: Residue Theory
Outline
6.1 The Residue Theorem
6.2 Trigonometric Integrals Over (0,2π)
6.3 Improper Integrals of Certain Functions Over (−∞,)
6.4 Improper Integrals Involving Trigonometric Functions
Ch.6: Residue Theory
6.1 The Residue Theorem
Introduction
In the previous chapters, we have seen how the theory of
contour integration lends great insight into the properties of
analytic functions
The goal this chapter is to explore another dividend of this
theory, namely, its usefulness in evaluating certain real
integrals
We shall begin by presenting a technique for evaluating
contour integrals that is known as residue theory
Then we will introduce some application of the theory to the
evaluating the real integrals
Ch.6: Residue Theory
6.1 The Residue Theorem
The Residue Theorem
If f(z)is analytic on and inside a simple closed positively
oriented contour Γexcept a single isolated singularity, z0,
lying interior to Γ,f(z)has a Laurent series expansion
f(z)=
j=−∞
aj(zz0)j
converging to some punctured neighborhood of z0
In particular, the above equation is valid for all zon the small
positively oriented circle Ccontinuously deformed from Γ(as
showninFig.6.1)
pf3
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Ch.6: Residue Theory

Chapter 6: Residue Theory

Li, Yongzhao

State Key Laboratory of Integrated Services Networks, Xidian University

June 7, 2009

Ch.6: Residue Theory

Outline

6.1 The Residue Theorem6.2 Trigonometric Integrals Over

π

6.3 Improper Integrals of Certain Functions Over

6.4 Improper Integrals Involving Trigonometric Functions

Ch.6: Residue Theory

6.1 The Residue Theorem

Introduction



In the previous chapters, we have seen how the theory ofcontour integration lends great insight into the properties ofanalytic functions



The goal this chapter is to explore another dividend of thistheory, namely, its usefulness in evaluating certain realintegrals



We shall begin by presenting a technique for evaluatingcontour integrals that is known as

residue theory



Then we will introduce some application of the theory to theevaluating the real integrals

Ch.6: Residue Theory

6.1 The Residue Theorem

The Residue Theorem



If

f

z

is analytic on and inside a simple closed positively

oriented contour

except a single isolated singularity,

z

0

lying interior to

f

z

has a Laurent series expansion

f

z

∞ ∑

j

=

−∞

a

j

z

z

0

j

converging to some punctured neighborhood of

z

0



In particular, the above equation is valid for all

z

on the small

positively oriented circle

C

continuously deformed from

(as

shown in Fig. 6.1)

Ch.6: Residue Theory

6.1 The Residue Theorem

The Residue Theorem (Cont’d)



According to the Continuous Deformation Invariance Theorem(page 231), we have

Γ

f

z

dz

C

f

z

dz



The last integral can be computed by termwise integration ofthe series along

C

. For all

j

the integral is zero, and for

j

we obtain the value

πia

1



Consequently we have

Γ

f

z

dz

πia

1

Ch.6: Residue Theory

6.1 The Residue Theorem

The Residue Theorem (Cont’d)



Thus the constant

a

1

plays an important role in contour

integration. Accordingly, we adopt the following terminology

Definition If

f

has an isolated singularity at the point

z

0

, then the coefficient

a

1

of

z

z

0

1

in the Laurent expansion for

f

around

z

0

is

called the

residue

of

f

at

z

0

and is denoted by

Res(

f

z

0

or

Res(

z

0

Ch.6: Residue Theory

6.1 The Residue Theorem

How to Compute the Residue



If

f

has a

removable singularity

at

z

0

, all the coefficients of

the negative powers of

z

z

0

in its Laurent expansion are

zero, and so, in particular, the residue at

z

0

is zero



If

f

has an

essential singularity

at

z

0

, we have to use its

Laurent expansion to find the residue at

z

0

(See Example 1 on

page 308)



If

f

has

a pole of order

m

at

z

0

, we have the following

theorem to find the residue

Theorem If

f

has a pole of order

m

at

z

0

, then

Res

f

z

0

) = lim

z

0

m

d

m

1

dz

m

1

[(

z

z

0

m

f

z

)]

Ch.6: Residue Theory

6.1 The Residue Theorem

How to Compute the Residue (Cont’d)



Example 2 gives us another way to compute the residue when f

is a rational polynomial



Let

f

z

P

z

/Q

z

, where the functions

P

z

and

Q

z

are both analytic at

z

0

and

Q

has a simple zero at

z

0

, while

P

z

0

. Then we have

Res(

f

z

0

P

z

0

Q

z

0

Ch.6: Residue Theory

6.3 Improper Integrals of Certain Functions Over

(

−∞

,

)

Improper Integrals of Certain Functions Over



Given any function

f

continuous on

, the limit

lim ρ

→∞

ρ

ρ

f

x

dx

is called the Cauchy principal value of the integral of

f

over

, and we write

p.v.

−∞

f

x

dx

:= lim

ρ

→∞

ρ

ρ

f

x

dx



We shall now show how the theory of residue can be used tocompute p.v. integrals for certain functions of

f



See Example 1 on page 319 to learn the basic idea of thealgorithm

Ch.6: Residue Theory

6.3 Improper Integrals of Certain Functions Over

(

−∞

,

)

Improper Integrals of Certain Functions Over

(Cont’d)

Lemma If

f

z

P

z

/Q

z

is the quotient of two polynomials such that

degree Q

degree P

then

lim ρ

→∞

C

  • ρ

f

z

dz

where

C

ρ

is the upper half-circle of radius

ρ

defined in Eq. (4) on

page 320 as shown in Figure 6.

Ch.6: Residue Theory

6.3 Improper Integrals of Certain Functions Over

(

−∞

,

)

Improper Integrals of Certain Functions Over

(Cont’d)



Then the improper integral

∞ −∞

f

x

dx

can be computed as

follows

p.v.

−∞

f

x

dx

= lim

ρ

→∞

πi

(residues inside Γ

ρ

Ch.6: Residue Theory

6.4 Improper Integrals Involving Trigonometric Functions

Improper Integrals Involving Trigonometric Functions



The purpose of this section is to use residue theory toevaluate integrals of the general forms:

p.v.

−∞

P

x

Q

x

cos

mx dx,

p.v.

−∞

P

x

Q

x

sin

mx dx



If we obtain the value of the integral

−∞

P

x

Q

x

e

imx

dx

the above two integrals can be obtained by computing the realand imaginary parts

Ch.6: Residue Theory

6.4 Improper Integrals Involving Trigonometric Functions

Improper Integrals Involving Trigonometric Functions(Cont’d)

Lemma If

m >

and

P/Q

is the quotient of two polynomials such that

degree Q

degree P

then

lim ρ

→∞

C

ρ

e

imx

P

x

Q

x

dz

where

C

  • ρ

is the upper half-circle of radius

ρ

Ch.6: Residue Theory

6.4 Improper Integrals Involving Trigonometric Functions

Improper Integrals Involving Trigonometric Functions(Cont’d)



Then the improper integral

∞ −∞

f

x

dx

can be computed as

follows

p.v.

−∞

e

imx

P

x

Q

x

dx

= lim

ρ

→∞

πi

(residues inside Γ

ρ



Thus

p.v.

−∞

cos

mx

P

x

Q

x

dx

p.v.

−∞

e

imx

P

x

Q

x

dx

p.v.

−∞

sin

mx

P

x

Q

x

dx

p.v.

−∞

e

imx

P

x

Q

x

dx