MA 412 Complex Analysis
Final Exam
Summer II Session, August 9, 2001.
1.
Find all the values of (
−
8
i
)
1
/
3
. Sketch the solutions.
1 pt.
Answer:
We start by writing
−
8
i
in polar form and then we’ll compute the
cubic root:
(
−
8
i
)
1
/
3
=
(8
e
−
iπ/
2
)
1
/
3
=
8
1
/
3
exp
(
i
(
−
π
6
+
2
πk
3
)
)
,
Hence
z
0
= 2 exp(
−
iπ/
6),
z
1
= 2 exp(
iπ/
2) = 2
i
,
z
2
= 2 exp(
i
7
π/
6).
2.
Suppose
v
is harmonic conjugated to
u
, and
u
is harmonic conjugated to
v
.
Show that
u
and
v
must be constant functions.
1 pt.
Answer
By definition,
v
is harmonic conjugated to
u
if Δ
u
= Δ
v
= 0 and
u
x
=
v
y
, u
y
=
−
v
x
hold. On the other hand, as
u
is harmonic conjugated to
v
,
we also have
v
x
=
u
y
and
v
y
=
−
u
x
. Then,
u
y
=
−
v
x
=
v
x
implies
v
x
= 0 and
u
y
= 0
and
u
x
=
v
y
=
−
v
y
implies
v
y
= 0 and
u
x
= 0
.
Hence,
u
(
x, y
) and
v
(
x, y
) are constant functions.
3.
Show that
f
(
z
) =

z

2
is differentiable at the point
z
0
= 0 but not at any
other point.
1 pt.
1
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z
2
z
0
z
1
2
= 2i
Figure 1: Roots of (
−
8
i
)
1
/
3
.
Answer:
We could show that
f
is not differentiable at any
z
0
∈
C
−
0 by
computing different limits for different trajectories. But we’ll use the necessary
conditions of differentiability (
i.e.
the CauchyRiemann equations) to do so.
First, we write
f
(
z
) =
u
(
x, y
)+
iv
(
x, y
), so
u
(
x, y
) =
x
2
+
y
2
and
v
(
x, y
) = 0.
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 Spring '08
 Grossman
 Complex Plane, Constant Functions

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