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Math 814 Exam 1
Due: Nov 7, 2007
You may consult books, homework solutions or the instructor, but no one else. All work
must be your own.
1.
Let
f
(
z
) =
z
2
+1
. Find equations for and sketch the curves obtained as images
of horizontal and vertical lines under
f
(
z
)
.
We have
f
=
u
+
iv
where
u
=
x
2

y
2
+1
and
v
= 2
xy
. First,
f
(
R
) = [1
,
∞
)
and
f
(
i
R
) = (
∞
,
1]
. If
x
=
a
6
= 0
is constant we have
u
= 1 +
a
2

y
2
,
v
= 2
ay,
so
u
= 1 +
a
2

v
2
4
a
2
.
In the
(
u, v
)
plane, this is a leftopening parabola with vertex
(1 +
a
2
,
0)
. As

a

increases, the vertex moves to the right and the parabola opens wider, becoming
more like a straight line.
If
y
=
b
6
= 0
is constant we get
u
= 1

b
2
+
v
2
4
b
2
.
In the
(
u, v
)
plane, this is a rightopening parabola with vertex
(1

b
2
,
0)
. As

b

increases, the vertex moves to the left and the parabola opens wider, becoming
more like a straight line. Each of these parabolas is perpendicular to all of the
previous ones.
2.
Repeat Problem 1 for
f
(
z
) = cos
z
.
We have
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 Three '09
 Cong
 Math, Equations

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