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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 1
Theme 8.
Taylor and Laurent Series.
Please revise the material on power series from 1styear Calculus.
Deﬁnition:
Let
{
z
0
,z
1
,...,z
n
,...
}
be a sequence of complex numbers. A series
∞
X
n
=0
z
n
=
z
0
+
z
1
+
z
2
+
···
converges
to
z
(we write
∑
∞
n
=0
z
n
=
z
) if the sequence of
partial sums
{
s
n
}
, where
s
n
=
z
0
+
z
1
+
z
2
+
···
+
z
n
, converges to
z
. If the sequence of partial sums diverges then the series
diverges
.
Lemma:
(1) If
z
n
=
x
n
+
iy
n
then
∑
z
n
converges to
z
=
x
+
iy
if and only if
∑
x
n
converges to
x
and
∑
y
n
converges to
y
.
(2) If
∑

z
n

converges then
∑
z
n
converges (and we say that
∑
z
n
converges absolutely
).
Example:
Consider the series
∑
∞
n
=0
z
n
=1+
z
+
z
2
+
z
3
+
···
, which is a geometric
sum with common ratio
z
.
The
n
th partial sum is
s
n
=1+
z
+
z
2
+
···
+
z
n
=
1

z
n
+1
1

z
If

z

<
1 then lim
n
→∞

z

n
+1
= 0, so lim
n
→∞
z
n
+1
= 0 as well and hence
∞
X
n
=0
z
n
= lim
n
→∞
s
n
= lim
n
→∞
1

z
n
+1
1

z
=
1
1

z
.
If

z

>
1 then lim
n
→∞
z
n
6
= 0, so the series diverges by the
n
thterm test.
If

z

= 1 then the series diverges if
z
=1
,

1
,i
or

i
and converges otherwise (use part (1)
of the Lemma).
Deﬁnition:
Let
z
0
∈
C
.
A series of the form
X
n
≥
0
a
n
(
z

z
0
)
n
, where
a
n
∈
C
, is called
a
power series about
z
0
.
1
Page 2
Theorem:
(Taylor’s Theorem) Let
z
0
∈
C
,
r
∈
R
with
r>
0, and
f
a function which
is analytic on
D
=
{
z
:

z

z
0

<r
}
. For each
z
∈
D
we have
f
(
z
)=
∞
X
n
=0
a
n
(
z

z
0
)
n
(
*
)
=
a
0
+
a
1
(
z

z
0
)+
a
z
(
z

z
0
)
2
+
···
+
a
n
(
z

z
0
)
n
+
···
,
where
a
n
=
f
(
n
)
(
z
0
)
n
!
.
That is, for each
z
∈
D
the series at (
*
) converges to the value
f
(
z
).
In words, this says that a Taylor series for a given complex function, expanded about the
point
z
0
will converge to the function at all points in the
largest open disc
centred at
z
0
in which
f
is
analytic.
The series at (
*
) is called the
Taylor series for
f
about
z
0
; if
z
0
= 0 we also call it the
Maclaurin series for
f
.
Deﬁnition:
For a Taylor series for
f
about
z
0
, the largest possible
R
for which it con
verges in

z

z
0

<R
is called the
radius of convergence
.
By Taylor’s Theorem the radius of convergence
R
is the distance from
z
0
to the nearest
singularity of
f
(or
R
=
∞
if
f
is entire), and the series diverges on

z

z
0

>R
.
Example:
State the region of convergence of the Taylor series for the function:
a.
f
(
z
)=
e
z
, about
z
=0
b.
f
(
z
)=
z
z

3
about
z
=0
c.
f
(
z
)=
z
z

3
about
z
= 1.
2
Page 3
Known Series:
In most of what follows, we will ﬁnd series by
manipulating
the standard Maclaurin Series
for the basic functions. You should learn these series carefully, if you do not already know
them.
e
z
=1+
z
+
z
2
2!
+
z
3
3!
+
...
=
∞
X
n
=0
z
n
n
!
.
sin
z
=
z

z
3
3!
+
z
5
5!

...
=
∞
X
n
=0
(

1)
n
z
2
n
+1
(2
n
+ 1)!
.
cos
z
=1

z
2
2!
+
z
4
4!

...
=
∞
X
n
=0
(

1)
n
z
2
n
(2
n
)!
.
sinh
z
=
z
+
z
3
3!
+
z
5
5!
+
...
=
∞
X
n
=0
z
2
n
+1
(2
n
+ 1)!
.
cosh
z
=1+
z
2
2!
+
z
4
4!
+
...
=
∞
X
n
=0
z
2
n
(2
n
)!
.
Since these functions are entire, the series converge for any value of
z
.
1
1

z
=1+
z
+
z
2
+
z
3
+
...
=
∞
X
n
=0
z
n
valid for

z

<
1
.
1
1+
z
=1

z
+
z
2

z
3
+
...
=
∞
X
n
=0
(

1)
n
z
n
valid for

z

<
1
.
Integrating we can write,
Log(1 +
z
)=
z

z
2
2
+
z
3
3

...
=
∞
X
n
=1
(

1)
n
+1
z
n
n
valid for

z

<
1
.
Example:
Find the Taylor Series for
f
(
z
)=
e
z
about
z
0
= 2.
3
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