Math 185. Sample Answers to Problem Set #9
Page 214
8.
Let
∞
X
n
=0
a
n
(
z

z
0
)
n
be the Taylor series for
f
(
x
) about
z
=
z
0
, and let
R
be its radius of convergence.
The assumption that
f
(
n
)
(
z
0
) = 0 for
n
≤
m
implies that
a
0
=
a
1
=
· · ·
=
a
m
= 0 ,
so
f
(
z
) =
∞
X
n
=
m
+1
a
n
(
z

z
0
)
n
= (
z

z
0
)
m
+1
∞
X
n
=0
a
n
+
m
+1
(
z

z
0
)
n
for all
z
with

z

z
0

< R
. (Here the last equality holds because the factor (
z

z
0
)
m
+1
is just a constant as far as the summation is concerned—see Exercise 7 on page 181.)
Therefore
g
(
z
) =
∞
X
n
=0
a
n
+
m
+1
(
z

z
0
)
n
for all
z
satisfying 0
<

z

z
0

< R
. This is also true when
z
=
z
0
since
g
(
z
0
) =
a
m
+1
=
f
(
m
+1)
(
z
0
)
(
m
+ 1)!
.
Therefore
g
(
z
) has a Taylor series at
z
0
, so it is analytic there.
10.
For all
N
∈
Z
+
and all
z
in the given annular domain, let
ρ
N
(
z
) =
S
2
(
z
)

N

1
X
n
=0
b
n
(
z

z
0
)
n
,
so that, as with (3) on page 207 (since 1
/
(
z

z
0
)
n
is also continuous on
C
),
Z
C
g
(
z
)
S
2
(
z
)
dz
=
N

1
X
n
=0
b
n
Z
C
g
(
z
)(
z

z
0
)

n
dz
+
Z
C
g
(
z
)
ρ
N
(
z
)
dz .
Next, we need to show that the given power series is uniformly convergent on
C
(and
this is the main difficulty in this part of the problem). Assume that our annular domain
is
R
1
<

z

z
0

< R
2
. Then the power series
T
(
w
) =
∞
X
n
=1
b
n
w
n
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
is convergent on the set

w

<
1
/R
1
since
T
(1
/
(
z

z
0
)) =
S
2
(
z
) , term for term. If
the contour
C
is given by
z
=
z
(
t
) ,
a
≤
t
≤
b
, then we can let
R
0
be the minimum
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Lim
 Power Series, Taylor Series, dz

Click to edit the document details