ENEE 241 02*
READING ASSIGNMENT 1
Tue 02/03 Lecture
Topic:
complex multiplication; complex exponentials
Textbook References:
section 1.3
Key Points:
•
Two complex numbers can be multiplied by expressing each number in the form
z
=
x
+
jy
,
then using distributivity and the rule
j
2
=

1 (i.e.,
j
is treated as the square root of

1).
•
The product of two complex numbers with polar coordinates (
r
1
, θ
1
) and (
r
2
, θ
2
) is the complex
number with polar coordinates (
r
1
r
2
, θ
1
+
θ
2
).
•
The product of a complex number
z
and its conjugate
z
*
equals the square modulus

z

2
.
•
If
z
has polar coordinates (
r, θ
), its inverse
z

1
has polar coordinates (
r

1
,

θ
).
•
The generic complex number
z
=
r
(cos
θ
+
j
sin
θ
) can be also written as
z
=
re
jθ
.
•
As is the case with real exponentials,
e
j
(
θ
+
φ
)
=
e
jθ
e
jφ
•
The functions cos
θ
and sin
θ
can be expressed in terms of complex exponentials:
cos
θ
=
e
jθ
+
e

jθ
2
and
sin
θ
=
e
jθ

e

jθ
2
j
Theory and Examples:
1
. The most common form for a complex number
z
incorporates the real and imaginary parts
as follows:
z
=
x
+
jy
This form, together with the convention that
j
×
j
=
j
2
=

1, allows us to multiply two
complex numbers together. For example,
(5

2
j
)(3

4
j
) = 15

20
j

6
j
+ 8
j
2
= 7

26
j
2
. In polar form, multiplication of complex numbers is simple. If
z
1
=
r
1
(cos
θ
1
+
j
sin
θ
1
)
and
z
2
=
r
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 Spring '08
 staff
 Exponential Function, Sin, Cos, Complex number, Euler's formula, ejθ

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