for
Chapter 8 Residue Theory
8.1 The Residue Theorem
Overview
We now have the necessary machinery to see
some amazing applications of the tools we developed in the last few
chapters. You will learn how Laurent expansions can give
useful information concerning seemingly unrelated properties of
complex functions. You will also learn how the ideas of
complex analysis make the solution of very complicated integrals of
real-valued functions as easy - literally - as the computation of
residues. We begin with a theorem relating residues to the
evaluation of complex integrals.
The Cauchy integral formulae in Section 6.5 are useful in evaluating contour integrals over a simple closed contour C where the integrand has the form and f is an analytic function. In this case, the singularity of the integrand is at worst a pole of order k at . We begin this section by extending this result to integrals that have a finite number of isolated singularities inside the contour C. This new method can be used in cases where the integrand has an essential singularity at and is an important extension of the previous method.
Definition 8.1 (Residue). Let
f(z) have a nonremovable isolated
singularity at the point . Then
f(z) has the Laurent series
representation for all z in some disk given
by
.
The coefficient of is
called the residue of f(z) at
and we use the notation
.
Example
8.1. If , then
the Laurent series of f about the point
has the form
, and
.
Example 8.2. Find if .
Solution. Using Example 7.7, we find that g(z)
has three Laurent series representations involving powers of
z. The Laurent series
valid in the punctured disk is
.
Computing the first few coefficients, we obtain
Therefore, .
Recall that, for a function f(z)
analytic in
and for any r with ,
the Laurent series coefficients of f(z)
are given by
(8-1) for ,
where
denotes the circle
with positive orientation. This gives us an important fact
concerning . If
we set in
Equation (8-1) and replace
with any positively oriented simple closed contour C
containing ,
provided
is the still only singularity of f(z)
that lies inside C, then we
obtain
(8-2) .
If we are able to find the Laurent series expansion for f(z),
then above equation gives us an important tool for evaluating contour
integrals.
Example 8.3. Evaluate where denotes the circle with positive orientation.
Solution. In Example 8.1 we showed that the residue
of at is . Using
Equation (8-2), we get
.
Theorem 8.1 (Cauchy's
Residue
Theorem). Let
D be a simply connected domain, and
let C be a simple closed
positively oriented contour that lies in D. If
f(z) is analytic
inside C and on C, except
at the points that
lie inside C, then
.
The situation is illustrated in Figure 8.1.
Figure 8.1 The domain D and contour C and the singular points in the statement of Cauchy's residue theorem.
Proof of Theorem 8.1 is in the book.
Complex
Analysis for Mathematics and Engineering
The calculation of a Laurent
series
expansion is tedious in most circumstances. Since the
residue at
involves only the coefficient
in the Laurent expansion, we seek a method to calculate the residue
from special information about the nature of the singularity at
.
If f(z) has a
removable singularity at ,
then for . Therefore, . Theorem
8.2 gives methods for evaluating residues at poles.
Theorem 8.2 (Residues at
Poles).
(i) If
f(z) has a simple pole
at , then .
(ii) If
f(z) has a pole of order 2
at , then .
(iii) If
f(z) has a pole of order 3
at , then .
(v) If
f(z) has a pole of order k
at , then .
Proof of Theorem 8.2 is in the book.
Complex
Analysis for Mathematics and Engineering
Example 8.4. Find the residue of at .
Solution. We write . Because has
a zero of order 3 at
and . Thus
f(z) has a pole of order 3
at . By
part (iii) of Theorem 8.2, we
have
This last limit involves an indeterminate form, which we evaluate
by using L'Hôpital's rule:
Example 8.5. Find where denotes the circle with positive orientation.
Solution. We write the integrand
as .
The singularities of f(z) that lie
inside
are simple poles at the points
and ,
and a pole of order 2 at the
origin. We compute the residues as
follows:
Finally, the residue theorem yields
The answer, , is
not at all obvious, and all the preceding calculations are required
to get it.
Example 8.6. Find where denotes the circle with positive orientation.
Solution. The singularities of the
integrand that
lie inside
are simple poles occurring at the points , as
the points , lie
outside . Factoring
the denominator is tedious, so we use a different
approach. If
is any one of the singularities of f(z)
, then we can use L'Hôpital's rule to compute :
Since , we
can simplify this expression further to yield
We now use the residue theorem to get
The theory of residues can be used to expand the quotient of two polynomials into its partial fraction representation.
Example 8.7. Let
P(z) be a polynomial of degree at
most 2. If a
, b and c
are distinct complex numbers, then
,
where
Solution. It will suffice to prove
that . We
expand f(z) in its Laurent series
about the point
by writing the three terms ,
,
and
in their Laurent series about the point
and adding them. The term
is itself a one-term Laurent series about the point . The
term
is analytic at the point ,
and its Laurent series is actually a Taylor series given
by
which is valid for .
Likewise, the Laurent expansion of the term
is
,
which is valid for . Thus
the Laurent series of f(z) about the
point
is
,
which is valid for ,
where . Therefore , and
calculation reveals that
Example 8.8. Express in partial fractions.
Solution. In Example 8.7 use and . Computing
the residues, we obtain
The formula for f(z) in Example 8.7
gives us
Remark 8.1. If a
repeated root occurs, then the process is similar, and it is easy to
show that if P(z) has degree of at
most 2, then
,
where
Example 8.9. Express in partial fractions.
Solution. Using the Remark 8.1
and and , we
have
where
Thus,
Extra Example 1. Express of in partial fractions.
Explore Solution for Extra Example 1.
Download This Mathematica Notebook
The Next Module is
Trigonometric Integrals via Contour Integrals
Return to the Complex Analysis Modules
Return to the Complex Analysis Project