


Complex Analysis 3.pdf
Complex_Analysis_3.pdf
Showing 912 out of 26
Complex Analysis 3.pdfTheme 8. Taylor and Laurent...
Complex_Analysis_3.pdfTheme 8. Taylor and Laurent Series.
Complex Analysis 3.pdfTheme 8. Tay...
Complex_Analysis_3.pdfTheme 8. Taylor and Laurent Series.
Page 9
Example:
Find the Laurent series for
f
(
z
)=
1
(
z

5)
2
(
z

4)
about 5 that converges when
z
= 7.
Lemma:
(Properties of power series – we have used many of these already)
1. (Uniqueness) If a series
∞
X
n
=
∞
c
n
(
z

z
0
)
n
converges to
f
(
z
) at all points
z
in some annular domain
D
about
z
0
, then it is the
Laurent series expansion for
f
about
z
0
.
2. If two power series converge on a domain
D
then we can add, subtract, multiply and
divide them as if we had polynomials, and the resulting series will converge as expected.
3. If a power series converges on a domain
D
then we can diﬀerentiate or integrate term
by term, and the resulting series will converge as expected.
9
Page 10
Theme 9.
Integrals and Laurent Series.
Now that we have constructed these rather strange series for complex functions, we ask the
question: What use are they? We have already seen that we can evaluate certain complex
integrals around closed contours, using CIF. But there are still many integrals we cannot do.
For example:
I

z

=
π
2
1
sin
z
dz.
The following remarkable theorem, tells us how to evaluate integrals using Laurent series. It
is based on the following observation:
Let
γ
be a simple closed contour around
z
0
, oriented anticlockwise. Then
Z
γ
(
z

z
0
)
n
dz
=
0
if
n
≥
0
2
πi
if
n
=

1
0
if
n
≤
2
by the Cauchy–Goursat Theorem (
n
≥
0) and Cauchy’s Integral formulae (
n
≤
1).
Theorem:
Suppose that
f
has one singularity at the point
z
0
, and that
f
has a Laurent
series about the point
z
=
z
0
of the form
f
(
z
)=
...
+
a

2
(
z

z
0
)
2
+
a

1
z

z
0
+
a
0
+
a
1
(
z

z
0
)+
...
If
γ
is any simple closed contour containing
z
0
, then
I
γ
f
(
z
)
dz
=2
πia

1
.
Thus, the value of the integral is controlled by the number
a

1
in the Laurent series for
f
,
when it is expanded about the singular point
z
0
.
Proof:
10
Page 11
The number
a

1
in the above theorem is called the
residue
if
f
at
z
0
. We use the notation:
a

1
= Res
z
=
z
0
f
(
z
)
.
Example:
Use Laurent series to ﬁnd
I

z

=3
e
z
z

1
dz.
Example:
Find the ﬁrst three nonzero terms of the Laurent series for
f
(
z
)=
1
sin
z
and
hence ﬁnd
I

z

=
π
2
1
sin
z
dz.
11
Page 12
Example:
Find
I

z

=2
sin
z
z
2

1
dz.
We now generalise the theorem relating integrals and residues to deal with functions which
have more than one singularity.
Theorem:
(Residue Theorem) Let
γ
be a simple closed contour oriented anticlockwise.
If
f
is analytic on and inside
γ
except for a ﬁnite number of isolated singularities
z
1
,...,z
n
inside
γ
, then
Z
γ
f
(
z
)
dz
=2
πi
n
X
k
=1
Res
z
=
z
k
f.
Proof:
By the CauchyGoursat Theorem,
Z
γ
f
(
z
)
dz
=
Z
γ
1
f
(
z
)
dz
+
Z
γ
2
f
(
z
)
dz
+
···
+
Z
γ
n
f
(
z
)
dz,
where the
γ
k
are nonintersecting circles around
z
k
, inside
γ
, oriented anticlockwise.
If
f
(
z
)=
∑
∞
n
=
∞
c
k
n
(
z

z
k
)
n
is the Laurent series for
f
about
z
k
, then
Z
γ
k
f
(
z
)
dz
=
∞
X
n
=
∞
Z
γ
k
c
k
n
(
z

z
k
)
n
=2
πic
k

1
=2
πi
Res
z
=
z
k
f.
12
Ace your assessments! Get Better Grades
Browse thousands of Study Materials & Solutions from your Favorite Schools
Pepperdine University
Pepperdine_University
School:
Mathematics_2A
Course:
Introducing Study Plan
Using AI Tools to Help you understand and remember your course concepts better and faster than any other resource.
Find the best videos to learn every concept in that course from Youtube and Tiktok without searching.
Save All Relavent Videos & Materials and access anytime and anywhere
Prepare Smart and Guarantee better grades
Students also viewed documents
lab 18.docx
lab_18.docx
Course
Course
3
Module5QuizSTA2023.d...
Module5QuizSTA2023.docx.docx
Course
Course
10
Week 7 Test Math302....
Week_7_Test_Math302.docx.docx
Course
Course
30
Chapter 1 Assigment ...
Chapter_1_Assigment_Questions.docx.docx
Course
Course
5
Week 4 tests.docx.do...
Week_4_tests.docx.docx
Course
Course
23
Week 6 tests.docx.do...
Week_6_tests.docx.docx
Course
Course
106