Download Residue Theory: Evaluating Real Integrals using Contour Integration and more Slides Mathematics in PDF only on Docsity!
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
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
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
