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This document provides an overview of a complex analysis class that covers poles, zeroes, residues, and Cauchy's Residue Theorem. It defines poles and zeroes, describes how to identify them for different functions, and gives a formula for computing residues at poles. Examples are provided to illustrate these concepts. Finally, Cauchy's Residue Theorem is stated and students are given exercises to evaluate contour integrals using this theorem by first finding the residues of singularities within the contour.

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145 views4 pages

23

This document provides an overview of a complex analysis class that covers poles, zeroes, residues, and Cauchy's Residue Theorem. It defines poles and zeroes, describes how to identify them for different functions, and gives a formula for computing residues at poles. Examples are provided to illustrate these concepts. Finally, Cauchy's Residue Theorem is stated and students are given exercises to evaluate contour integrals using this theorem by first finding the residues of singularities within the contour.

Uploaded by

Suka Joshua
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Complex Analysis

Math 214 Spring 2014 Fowler 307 MWF 3:00pm - 3:55pm


c
2014 Ron Buckmire http://faculty.oxy.edu/ron/math/312/14/

Class 23: Monday March 31


TITLE Poles, Zeroes and Residues
CURRENT READING Zill & Shanahan, §6.4-6.5
HOMEWORK Zill & Shanahan, §6.4 2,6,21. §6.5 2,12,17,23.
SUMMARY
We shall be introduced to the concept of residues, and we shall learn about Cauchy’s Residue
Theorem.
Zeroes and Poles
So, far we have had a lot of experience finding the poles of a function and this was important
in evaluating contour integrals. The problem of finding a pole is equivalent to finding the
zero of a related function. Let’s formalize these definitions:
DEFINITION: Zero
A point z0 is called a zero of order m for the function f (z) if f is analytic at z0 and f and
its first m − 1 derivatives vanish at z0, but f (m) (z0 ) 6= 0.

DEFINITION: Pole
A point z0 is called a pole of order m of f (z) if 1/f has a zero of order m at z0 .

Identifying Poles and Zeroes


Let f be analytic. Then f has a zero of order m at z0 if and only if f (z) can be written
as f (z) = g(z)(z − z0 )m where g is analytic at z0 and g(z0 ) 6= 0.

g(z)
If f (z) can be written as f (z) = where g(z) is analytic at z0, then f has a pole
(z − z0)m
of order m at z = z0 and g(z0) 6= 0

How do we find the poles of a function? Well, if we have a quotient function f (z) = p(z)/q(z)
where p(z)are analytic at z0 and p(z0 ) 6= 0 then f (z) has a pole of order m if and only if
q(z) has a zero of order m.
3z + 2
EXAMPLE We will classify all the singularities of f (z) = 4 . How many singularities
z + z2
does f (z) have? And of what order?

1
Complex Analysis Worksheet 23 Math 312 Spring 2014

Groupwork
Let’s try and classify all the singularities of the following functions:
4
(a) A(z) =
z 2 (z − 1)3

sin z
(b) B(z) =
z2 − 4

(c) C(z) = tan z

z
(d) D(z) =
z2 − 6z + 10

Residues
Once we know all the singularities of a function it is useful to compute the residues of that
function. If a function f (z) has a pole of order m at z0, the residue, denoted by Res(f ; z0)
or Res(z0) is given by the formula below:
 
1 dm−1 m
Res(f ; z0 ) = lim [(z − z0) f (z)]
z→z0 (m − 1)! dz m−1
EXAMPLE
3z + 2
Let’s find the residues of the singularities of f (z) = .
z4 + z2

2
Complex Analysis Worksheet 23 Math 312 Spring 2014

Exercise
Find the residues of all the singularities we previously classified for the following functions.
What is z0 and m in each case?
4
(a) A(z) = 2
z (z − 1)3

sin z
(b) B(z) =
z2 − 4

(c) C(z) = tan z

z
(d) D(z) =
z2 − 6z + 10

3
Complex Analysis Worksheet 23 Math 312 Spring 2014

Cauchy’s Residue Theorem


If f is analytic on a simple (positively oriented) closed contour Γ and everywhere inside Γ
except the finite number of points z1 , z2, · · · zn inside Γ, then
I X
n
f (z) dz = 2πi Res(f ; zk )
Γ k=1

EXAMPLE I
3z + 2
Let’s use the CRT to evaluate the following dz
|z|=2 z 2 (z 2 + 1)

sc GroupWork
Use Cauchy’s Residue Theorem (CRT) to evaluate the following integrals:
I
4
(a) 2 3
dz
|z|=5 z (z − 1)

I
sin z
(b) dz
|z|=5π z2 − 4

I
(c) tan z dz
|z|=2π

I
z
(d) dz
|z|=8 z 2 − 6z + 10

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