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principal
argument
of
z
and is denoted by
Arg
( )
z
. This rectangular–polar relationship is illustrated in
Figure
10
(
Complex plane
).
IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Mathematics I
Semester 1, 2021/2022
Chapter I:
Complex Numbers
Lecturer
Associate Professor Dr. Abdurahim Okhunov
19
Definition 6
(
General Polar form of a Complex Number
)
:
For
k
any integer,
i
k
2
cos
si
2
2
i
i
y
z
x
n
r
k
k
r
e
(4)
The number
r
is called the
modulus
or
modulus value
of
z
and is denoted by mod
z
or
z
.
The polar angle that the line joining
z
to the origin makes with the polar axis is called the
argument
of
z
and is denoted by
arg
z
. from
Figure 8
we see the following relationships:
2
2
r
x
y
,
0
0
180
2
360
,
.
,
k
k
k
z
For example:
,
2
2
z
x
y
Never
mo
negetive
z
d
r
,
,
arg
k
k
any in eg
2
er
z
t
,
where
sin
y
r
=
and
cos
x
r
=
. The argument
is usually chosen so that
0
0
180
180
<
or
<
.
Example 12
(
From Rectangular to Polar
)
:
Write parts
A
–
C
in polar form,
in radians,
<
. Compute the modulus and
arguments for parts
A
and
B
exactly; compute the modulus and argument for part
C
to two decimal
places.
,
,
1
2
3
A
1
3
5
2
i
i
)
B)
C
z
z
i
z
)
.
Solution:
Locate in a complex plane first; then if
x
and
y
are associated
with special angles,
r
and
can often be determined by inspection.
A)
A sketch shows that
1
z
is associated with a special
0
45
triangle
(
as
in
Figure
).
Thus,
by
inspection.
4
4
n
7
ot
2
r
,
, and
x
0
i
y
co
z
=
+
i
i
s
s
=
x
r
r
in
e
y
x
y
r
2
2
r
x
y
2
2
r
b
a
sin
b
r
cos
a
r
y
x
( , )
a
b
0, 0
Figure 10
1
r
1
-
i
-
1
x
y
IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Mathematics I
Semester 1, 2021/2022
Chapter I:
Complex Numbers
Lecturer
Associate Professor Dr. Abdurahim Okhunov
20
1
i
cos
sin
2
4
4
z
+
.
B)
A sketch shows that
2
z
is associated with a special
0
0
90
60
triangle (
as in Figure
). Thus, by inspection.
6
2
5
,
r
, and
2
5
5
5
2
6
6
i
co
z
s
sin
i
2
6
e
.
Example 13
(
From Rectangular to Polar
)
:
Write the following complex numbers in polar form using their principal arguments:
a)
1
z
i
1
,
b)
2
z
i
1
3
.
Solution:
a)
The first we will compute
r
and
:
1
2
2
1
1
2
r
z
and
tan
1
1
a
b
1
.
Therefore,
0
45
z
Arg
4
and by
equation
(
1
) we have
4
1
i
cos
si
2
n
cos
sin
r
e
2
4
i
z
4
i
b)
Here again we will compute
r
and
:
2
2
2
1
3
1
z
4
2
r
3
and
tan
1
b
3
a
3
.
Since
2
z
lies in the fourth quadrant, we must have,
0
60
rg
z
3
A
and by
equation
(
2
) we have
1
i
3
cos
si
2
n
cos
s
r
i
i
2
3
n
e
i
z
3
=
As shown in
Figure 11
.
3
Im
Re
2
1
1
0
4
3
2
2
1
1
1
i
z
2
1
3
i
z
Figure 11
1
r
3
i
3
x
y
IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Mathematics I
Semester 1, 2021/2022
Chapter I:
Complex Numbers
Lecturer
Associate Professor Dr. Abdurahim Okhunov
21
Multiplication and Division in Polar form:
The following important result is known as
Euler’s formula
:
cos
sin
i
r e
i
z
r
,
(5)
Every non–zero complex number
z
a b
i
, with polar coordinates
,
r
can be
written as
cos
sin
a b
r
z
i
i
in polar form, and
i
i
b
re
z
a
in exponential form.
Some people call both of the above forms the polar form of
z
, since they are
both based on the polar coordinates of
z
.
Theorem 1
(
Products and Quotients in Polar Form
)
:
If
1
1
1
r
z
e
i
and
2
2
2
r
z
e
i
, then:
1
2
1
2
1
1
2
2
1.
2.
.
1
2
1
2
1
2
1
1
1
2
2
2
z
z
z
z
r e
r e
r r e
r e
r
e
r e
r
i
i
i
i
i
i
(5)
Example 14
(Products and Quotients)
:
If
0
1
45
8
e
z
i
and
0
2
30
2
e
z
i
, then find
1.
2.
.
1
1
2
2
z
z
z
z
Solution:
1)
0
0
1
2
0
0
0
45
30
45
30
75
8
2
8 2
16
z
z
e
e
e
e
i
i
i
i
2)
0
1
0
2
0
0
0
45
30
45
30
15
8
2
8
2
z
4
z
e
e
e
e
i
i
i
i
.
1.3
Statement Euler’s formula. De Moivre’s Theorem for a rational index
Trigonometric Representations of
z
Euler's formula was proved (in an obscured form) for the first time by Roger Cotes in 1714,
then rediscovered and popularized by Euler in 1748.
IIUM, Faculty of Engineering,
Department Engineering in Science
Engineering Mathematics I
Semester 1, 2021/2022
Chapter I:
Complex Numbers
Lecturer
Associate Professor Dr. Abdurahim Okhunov
22
Euler proved this formula using power series expansions of exponential, sine and cosine
functions (and this proof can be subject of your project). This formula allows the following
simplification
i
e
i
=
= cos +
sin =
cos
i
+ sin
r
r
r
x
y
i
i
=
z
r e
.
(6)
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