


Complex Analysis 3.pdf
Complex_Analysis_3.pdf
Showing 48 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 4
Example:
Let
g
(
z
)=
z
2
e
5
z
. Find the Taylor series for
g
about
z
0
= 0.
Example:
Find the Taylor series for sin
z
about
z
0
=
π
Example:
Let
h
(
z
)=
10
(
z

i
)(
z
+ 3)
. Find the Taylor series for
h
about
z
0
= 2, that is,
in powers of (
z

2).
What is its radius of convergence?
We have the partial fractions
decomposition
h
(
z
)=
10
(
z

i
)(
z
+ 3)
=
3

i
z

i
+
i

3
z
+3
,
4
Page 5
Example:
What is the radius of convergence for the Taylor series for sec
z
about 0? Find
the ﬁrst three nonzero terms of this series.
5
Page 6
We can now deduce a version of L’Hˆ
opital’s Rule.
Theorem:
(L’Hˆ
opital’s Rule) If
f
and
g
are analytic at
z
0
and
f
(
z
0
)=0=
g
(
z
0
) = 0 but
g
0
(
z
0
)
6
= 0, then
lim
z
→
z
0
f
(
z
)
g
(
z
)
= lim
z
→
z
0
f
0
(
z
)
g
0
(
z
)
.
Since
f
and
g
are analytic at
z
0
, Taylor’s Theorem gives
f
(
z
)=
∞
X
n
=0
a
n
(
z

z
0
)
n
and
g
(
z
)=
∞
X
n
=0
b
n
(
z

z
0
)
n
for
z
near
z
0
. Since
a
0
=
f
(
z
0
) = 0 and
b
0
=
g
(
z
0
) = 0 we have
f
(
z
)=
a
1
(
z

z
0
)+
a
2
(
z

z
0
)
2
+
a
3
(
z

z
0
)
3
+
···
g
(
z
)=
b
1
(
z

z
0
)+
b
2
(
z

z
0
)
2
+
b
3
(
z

z
0
)
3
+
···
for
z
near
z
0
.
Note that
g
0
(
z
0
)=
b
1
6
= 0. Now
lim
z
→
z
0
f
(
z
)
g
(
z
)
= lim
z
→
z
0
a
1
(
z

z
0
)+
a
2
(
z

z
0
)
2
+
a
3
(
z

z
0
)
3
+
···
b
1
(
z

z
0
)+
b
2
(
z

z
0
)
2
+
b
3
(
z

z
0
)
3
+
···
= lim
z
→
z
0
a
1
+
a
2
(
z

z
0
)+
a
3
(
z

z
0
)
2
+
···
b
1
+
b
2
(
z

z
0
)+
b
3
(
z

z
0
)
2
+
···
=
a
1
b
1
= lim
z
→
z
0
f
0
(
z
)
g
0
(
z
)
.
6
Page 7
Laurent Series
If a function
f
fails to be analytic at a point
z
0
then we cannot ﬁnd a Taylor series converging
to
f
in any neigbourhood of
z
0
, but it is often possible to ﬁnd a series representation for
f
involving both positive and negative powers of
z

z
0
.
Example:
Consider the function
f
(
z
)=
e
z
z
2
.
Theorem:
(Laurent’s Theorem) Suppose that a function
f
is analytic throughout the an
nulus
D
=
{
z
:
R
1
<

z

z
0

<R
2
}
. Let
γ
be any simple closed contour around
z
0
, oriented
anticlockwise, lying entirely in
D
. For each
z
∈
D
we have
f
(
z
)=
···
+
b
2
(
z

z
0
)
2
+
b
1
z

z
0
+
a
0
+
a
1
(
z

z
0
)+
a
2
(
z

z
0
)
2
+
···
(
†
)
where
a
n
=
1
2
πi
Z
γ
f
(
z
)
(
z

z
0
)
n
+1
dz
for
n
=0
,
1
,
2
···
b
n
=
1
2
πi
Z
γ
f
(
z
)
(
z

z
0
)

n
+1
dz
for
n
=0
,
1
,
2
···
We call (
†
) the
Laurent series for
f
about
z
0
.
In practice, we hardly ever use the for
mulas for the coeﬃcients
a
n
and
b
n
– we manipulate known series instead (just as we do to
ﬁnd Taylor series).
If
f
is analytic on the disc

z

z
0

<R
2
then
b
n
=
1
2
πi
Z
γ
f
(
z
)(
z

z
0
)
n

1
dz
=0
for
n
=0
,
1
,
2
···
,
7
Page 8
by the CauchyGoursat Theorem and (
†
) reduces to a Taylor series about
z
0
. So a Taylor
series is a Laurent series (but not vice versa).
Example:
Find the Laurent series for
f
(
z
)=
e
1
z
about 0.
Example:
Let
f
be the function
f
(
z
)=
1
(
z

1)(
z

4)
=

1
/
3
z

1
+
1
/
3
z

4
.
a) Find the largest open annuli or discs centred at
i
in which
f
is analytic;
b) ﬁnd the Laurent series for
f
about
i
which is valid when
z
= 2;
8
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