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Complex
Analysis. FreeTutorial
8.1
The modulus and argument of the product
8.2
Multiplication of complex numbers as stretching
(squeezing) and rotation
8.2.1
Multiplying a complex number by a real
number
8.2.2
Multiplying a complex number by imaginary
unit i
8.2.3
Multiplying a complex number by -i
8.2.4
Multiplying a complex number by positive
integer powers of i
8.2.5
Multiplying a complex number by negative
integer powers of i
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8.1 The modulus and argument of the product
Let us consider two complex
numbers z1
and z2
in a polar form.
z1
= r1(cos +i
sin),
z2
= r2(cos +i
sin).
Their product can be written in the form
z1z2
=r1r2[(coscos
- sinsin)
+
+ i(sincos
+ cossin)].
By means of the addition theorems of the
sine and cosine this expression can be
simplified to
z1z2
=
r1r2[(cos(+)
+ i(sin(+)].
(1.18)
The product z1z2
has the modulus r1r2
and the argument +.
Figure 1.15 illustrates the multiplication
of the complex numbers. The length of
the z1z2
vector equals the product of the lengths
of z1
and z2.
r1
= |z1|,
=
Arg(z1),
r2
= |z2|,
=
Arg(z2).
|z1
z2|
= |z1|
|z2|,
(1.19)
Arg(z1z2)
= Arg(z1)
+ Arg(z2).
(1.20)
The formula (1.20) means
that the set of values of Arg(z1z2)
is obtained by forming all possible sums
of value of Arg(z1)
and Arg(z2).
Figure 1.15 The geometry of multiplication.
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8.2
Multiplication of complex numbers as stretching
(squeezing) and rotation
The geometric interpretation
of multiplication of complex numbers z1z2
is stretching (or squeezing) and rotation
of vectors in the plane. If you have two
complex numbers z1
and z2
(z1
0, z2
0),
you can draw a vector z1,
multiply its length by the |z2|,
and rotate the resulting vector counterclockwise
through the angle Arg(z2).
If |z2|
> 0, we deal
with stretching. If |z2|
< 0, it is a
case of squeezing.
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8.2.1
Multiplying a complex number by a real
number
Let us multiply a complex
number z
= x
+
yi
by the real number a.
(x
+
yi)a
= xa
+ yai.
The multiplication of a complex number
by the real number a,
is a transformation which stretches the
vector by a factor of a
without rotation.
Let us consider two cases: a
= 2, a
= 1/2.
Geometrically, when we double a complex
number, we double the distance from the
origin, to the point in the plane. When
we multiply a complex number by 1/2,
the result will be half way between the
origin and z.
Multiplication by a
> 0
is a transformation which stretches the
complex plane C
by a factor of a
away from the origin.
Multiplication by a
< 0
is a transformation which squeezes C
toward 0.
Top
8.2.2
Multiplying a complex number by imaginary
unit i
zi
= (x
+
yi)i
= - y
+ xi
.
The point z
in C is located
x
units to the right of the imaginary axis
and y
units above the real axis (if x
> 0, y
> 0).
The point zi
is located y
units to the left, and x
units above.
Multiplying by i
has rotated a point z
90° counterclockwise around the origin
to the point zi.
See Figure 1.16
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8.2.3
Multiplying a complex number by -i
z(-i)
= (x
+
yi)(-i)
= y
- xi.
Multiplication by -i
gives a 90° clockwise rotation around
the origin or a 270° counterclockwise
rotation around the origin.
Top
8.2.4
Multiplying a complex number by positive
integer powers of i
Multiplying z
by i
means 90° counterclockwise rotation
of a point z
around the origin to the point zi.
Multiplying z
by i2
means 180° counterclockwise rotation
of a point z
around the origin to the point zi2
or 90° counterclockwise rotation of
a point zi
around the origin to the point zi2.
Multiplying z
by i3
means 270° counterclockwise rotation
of a point z
around the origin to the point zi3
or 90° counterclockwise rotation of
a point zi2
around the origin to the point zi3.
Multiplying z
by i4
means 360° counterclockwise rotation
of a point z
around the origin. The point z
corresponds to z,
zi4,
zi8,
and so on. The circle is repeated.
Figure 1.16 shows the multiplication
by powers of i.
The point z
corresponds to z,
zi4,
zi8,
zi12,
… . The point zi
corresponds to zi,
zi5,
zi9,
zi13
and so on.
Figure 1.16 Multiplication
by i
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8.2.5
Multiplying a complex number by negative
integer powers of i
Multiplying z
by i
-1 means 90°
clockwise rotation of a point z
around the origin to the point z(i)-1.
Multiplying z
by i
-2
means 180° clockwise rotation of a
point z
around the origin to the point z(i)-2.
The point z(i)-2
also corresponds to 90° clockwise
rotation of a point z(i)-1
around the origin.
Multiplying z
by i
-3 means
270° clockwise rotation of a point
z
around the origin to the point z(i)-3.
The point z(i)-3
also corresponds to 90° clockwise
rotation of a point z(i)-2
around the origin.
Multiplying z
by i
-4 means
360° counterclockwise rotation of
a point z
around the origin. The point z
corresponds to z,
zi
-4,
zi
-8,
and so on. The circle is repeated.
Figure 1.17 shows the multiplication
by negative powers of i.
Figure 1.17 Multiplication by negative
powers of i
by
Tetyana Butler
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