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ASSIGNMENT 1
MATH 270, WINTER 2010.
DUE: 5PM, JANUARY 26, 2010.
Assignments must have your name, student number, course number and TA’s name on the
ﬁrst page. Please submit them by placing them in the box marked “Assignments” outside the
Maths Ofﬁce (Burnside Hall, 10th ﬂoor). Show all working and justiﬁcations!
(1) (a) Let
z
=
2
+
3
i
and
w
=
3

2
i
. Evaluate the following in the form
a
+
b
i
,
a
,
b
∈
R
:
(i)
z
+
w
(ii)
z
*

3
w
(iii)
z
2
+
2
zw
+
w
2
(iv)
(
z
+
w
)
/
w
*
(v)
z
/

z

(vi)
w
*
z

z
*
w
[
3 marks
]
(b) Convert the following to polar form:
(i) 3

3
i
(ii)

√
3
+
3
i
[
2 marks
]
(c) Find all solutions of
z
5
=
1 and draw them on the complex plane. Let
ω
be the solution
with the smallest
positive
argument. Show that:
ω
*
=
ω

1
and
ω
2
+
ω
+
1
+
ω

1
+
ω

2
=
0
.
Hence show that
z
=
ω
+
ω

1
satisﬁes
z
2
+
z

1
=
0
and solve for
z
(
hint: your drawing should allow you to decide if z
>
0
or z
<
0).
Conclude that
cos
2
π
5
=
√
5

1
4
and
sin
2
π
5
=
p
10
+
2
√
5
4
.
[
5 marks
]
[
Total: 10 marks
]
(2) Consider the function
f
(
z
) =
arctan
(
i
z
)
where
z
∈
C
and
f
takes values in
C
.
(a) Differentiate
f
(
z
)
with respect to
z
.
[
1 mark
]
(b) Use partial fractions to reintegrate.
[
3 marks
]
(c) Show that
arctan
(
i
z
) =
i
2
ln
1
+
z
1

z
.
(1)
[
2 marks
]
(d) Find the real values of
z
of smallest absolute value for which
f
(
z
)
is undeﬁned.
[
1 mark
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 Winter '10
 Ridout
 Math

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