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Polar form of a Complex number
Example:
Let
z
= 1 +
i
. Then Polar form of
z
=

z

e
iθ
. For
z
= 1 +
i,

z

=
√
1
2
+ 1
2
=
√
2
and
θ
=
π/
4
. So
z
=
√
2
e
i
π
4
Multiplication:
Let
z
1
=

z
1

e
iθ
1
,z
2
=

z
2

e
iθ
2
then
z
1
z
2
=

z
1

z
2

e
i
(
θ
1
+
θ
2
)
=

z
1

z
2

(cos (
θ
1
+
θ
2
) +
i
sin (
θ
1
+
θ
2
))
Division:
Let
z
1
=

z
1

e
iθ
1
,z
2
=

z
2

e
iθ
2
then
z
1
z
2
=

z
1


z
2

e
i
(
θ
1

θ
2
)
=

z
1


z
2

(cos (
θ
1

θ
2
) +
i
sin (
θ
1

θ
2
))
Powers:
Let
z
=

z

e
iθ
then
z
n
=

z

n
e
i
(
nθ
)
=

z

n
(cos (
nθ
) +
i
sin (
nθ
))
Example:
Let z=1+i, ﬁnd
z
4
.
Solution:
We have
z
=
√
2
e
i
π
4
. Then
z
4
= (
√
2)
4
e
iπ
= 4(cos (
π
) +
i
sin (
π
)) =

4
nth roots of a complex number
. Let
z
=

z

e
iθ
. Then
z
1
n
=

z

1
n
e
i
(
θ
+2
πk
n
)
=

z

1
n
(cos (
θ
+ 2
πk
n
) +
i
sin (
θ
+ 2
πk
n
))
where
k
= 0
,
1
,...,n
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 Spring '11

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