(Write the integrals to compute the coefficients in the final series expansions without
attempting to evaluate them).
4.
Draw pictures of the following subsets of the complex plane (give as much detail
and be as accurate as possible):
(a)
{
z
:
z
= ¯
z
}
(b)
{
z
:
Im
z
> (
Re
z
)
2
}
5.
For each of the following sums say whether it is absolutely convergent or not, and
give a justification for your answer:
(a)
∞
∑
n
=
1
parenleftBig
2
2
+
i
parenrightBig
n
(b)
∞
∑
n
=
1
n
−
i
n
+
i
6.
Give the radius of convergence of the following power series, justify your answer.
(a)
∞
∑
n
=
1
n
4
z
n
(b)
∞
∑
n
=
1
z
n
(
n
!
)
2
7.
Evaluate the following contour integral
contintegraldisplay

z
=
4
dz
z
2
−
4
−
contintegraldisplay

z
+
2
=
1
dz
z
2
−
4
(2)
8.
Evaluate the following contour integral
contintegraldisplay

z
=
1
(
2
z
2
+
3
¯
z
)
dz
.
(3)
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9.
For the following function, locate and classify all the singularities as a pole of some
finite order, essential singularity, or removeable singularity. Compute the residues
at the poles of finite order:
f
(
z
)
=
e
i
z
z
+
1
z
(
z
−
2
)
2
.
(4)
10.
For the following functions give the radius of convergence of the Taylor series cen
tered at the indicated point
(a)
f
(
z
)
=
e
z
−
1
z
centered as
z
0
=
1
.
(b)
f
(
z
)
=
(
z
+
1
)(
z
−
2
)
(
z
−
2
i
)
2
centered at
z
0
=
2
.
11.
For the function
f
(
z
)
=
1
z
(
z
−
4
)
,
find the Laurent expansions valid in
(a) 0
<

z

<
4
(b) 4
<

z

(c) 0
<

z
−
4

<
4
(d) 4
<

z
−
4

Extra Credit: Suppose that
f
(
z
)
is an analytic function in the disk
{
z
: 
z
−
1

<
2
}
,
and that
f
[
j
]
(
1
)
=
0
,
for
j
=
0
,
1
,
2
, . . .
What can we conclude about
f
(
z
)
in this disk?
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 Spring '08
 STAFF
 Calculus, Power Series, Taylor Series, Sets, thermal contact, stationary heat equation

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