136
INTRODUCTION TO COMPLEX VARIABLES
Y
imag
z
--.
-
real
X
Figure
6.1
A
Plot
of
the Complex Number
z
=
x
+
iy
in
the Complex Plane
complex number is plotted on the horizontal
axis
and the imaginary part is plotted
on
the vertical axis.
A
plot of the complex number
=
x
+
iy
is shown
in
Figure 6.1.
The
magnitude
of
the complex number
z
=
x
+
iy
is
1g1
=
dW,
(6.3)
which
is
simply the distance from the origin to the point on the complex plane.
6.1.2
Complex
Conjugates
The conjugate of a complex number can be viewed
as
the reflection
of
the number
through the real axis
of
the complex plane.
If
g
=
x
+
iy,
its complex conjugate
is
defined
as
z*
3
x
-
iy.
(6.4)
The complex conjugate is useful for determining the magnitude of
a
complex
number:
(6.5)
2
zz*
=
(x
+
iy)(x
-
iy)
=
x2
+
y2
=
JzJ
.
In
addition, the real and imaginary
parts
of
a complex variable can be isolated using
the pair
of
expressions:
g
+
g*
-
(x
+
iy)
+
(x
-
iy)
-
-x
--
2
2
g
-
g*
-
(x
+
iy)
-
(x
-
iy)
=
y.
--
2i 2i
6.1.3 The Exponential Function
and
Polar
Representation
There is another representation of complex quantities which follows from Euler’s
equation,
cos
0
+
i
sin
0.
(6.8)
,iO
=
To understand
this
expression, consider the Taylor series expansion of
ex
around zero:
