Residue Theorem - Residue Calculus
Chapter 8 Residue Theory
8.1 The Residue Theorem
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
The coefficient of is called the residue of f(z) at and we use the notation
8.1. If , then
the Laurent series of f about the point
has the form
Explore Solution 8.1.
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
Explore Solution 8.2.
Recall that, for a function f(z)
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
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
Explore Solution 8.3.
Theorem 8.1 (Cauchy's
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
expansion is tedious in most circumstances. Since the
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
(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
This last limit involves an indeterminate form, which we evaluate
by using L'Hôpital's rule:
Explore Solution 8.4.
Example 8.5. Find where denotes the circle with positive orientation.
Solution. We write the integrand
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.
Explore Solution 8.5.
Example 8.6. Find where denotes the circle with positive orientation.
Solution. The singularities of the
are simple poles occurring at the points , as
the points , lie
outside . Factoring
the denominator is tedious, so we use a different
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
Explore Solution 8.6.
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
Solution. It will suffice to prove
that . We
expand f(z) in its Laurent series
about the point
by writing the three terms ,
in their Laurent series about the point
and adding them. The term
is itself a one-term Laurent series about the point . The
is analytic at the point ,
and its Laurent series is actually a Taylor series given
which is valid for .
Likewise, the Laurent expansion of the term
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
Explore Solution 8.7.
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
Explore Solution 8.8.
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
Example 8.9. Express in partial fractions.
Solution. Using the Remark 8.1
and and , we
Explore Solution 8.9.
Extra Example 1. Express of in partial fractions.
Explore Solution for Extra Example 1.
Exercises for Section 8.1. The Residue Theorem
The Next Module is
Trigonometric Integrals via Contour Integrals
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This material is coordinated with our book Complex Analysis for Mathematics and Engineering.
(c) 2012 John H. Mathews, Russell W. Howell