Complex
Analysis. FreeTutorial
13.
Integer powers of complex numbers
13.1 Modulus z<1
13.2 Modulus z>1
13.3 Modulus z=1
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13. Integer powers of complex numbers
Integer powers of complex
numbers are just special cases of products.
The n
th power of z,
written z^{n},
is equal to
z^{n}
= r^{n}^{}(cos(n)+i
sin(n)),
(1.24)
where n
is a positive or negative integer or zero.
If we know a complex number z,
we can find z^{n}.
The modulus z^{}^{n}
is the n^{th}
power of the modulus of z,
the argument of z^{n}
is n
times the argument of z.
There are three cases: z^{}^{}<1,
z^{}>1,
z^{}^{n}=1.
Let us consifer all of them.
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13.1
Integer powers of complex numbers. z<1
If z^{}^{}<1,
then z^{}^{n}<z<1
for any integer n.
The bigger n
the less z^{}^{n}.
Example:
z^{}=
0.9; arg(z)
= 30°. Find z^{2},
z^{3},
z^{4},
z^{5},
z^{6}.
z^{2}=
0.81; arg(z)
= 60°;
z^{2}=
0.73; arg(z)
= 90°;
z^{2}=
0.66; arg(z)
= 120°;
z^{2}=
0.59; arg(z)
= 150°;
z^{2}=
0.53; arg(z)
= 180°.
Each higher power is 30°
further along and closer to 0.
Six powers are displayed as points on
a spiral in the figure 1.25. This spiral
is called an exponential sprial.
Figure
1.25. An exponential sprial. z^{}^{}<1.
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13.2
Integer powers of complex numbers. z>1
If z^{}^{}>1,
then z^{}^{n}>z>1
for any integer n.
The bigger n
the bigger z^{}^{n}.
Example:
z^{}=
1.2; arg(z)
= 30°. Find z^{2},
z^{3},
z^{4},
z^{5},
z^{6}.
z^{2}=
2.4; arg(z)
= 60°;
z^{2}=
2.9; arg(z)
= 90°;
z^{2}=
3.5; arg(z)
= 120°;
z^{2}=
4.2; arg(z)
= 150°;
z^{2}=
5; arg(z)
= 180°.
Six powers are displayed
as points on a spiral in the figure 1.26.
Figure
1.26. An exponential sprial. z
>1.
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13.3
Integer powers of complex numbers. z=1
If z^{}^{}=1,
then z^{}^{n}=1
for any integer n.
Example:
z^{}=
1; arg(z)
= 30°. Find z^{2},
z^{3},
z^{4},
z^{5},
z^{6}.
z^{2}=
1; arg(z)
= 60°;
z^{2}=
1; arg(z)
= 90°;
z^{2}=
1; arg(z)
= 120°;
z^{2}=
1; arg(z)
= 150°;
z^{2}=
1; arg(z)
= 180°.
Figure
1.27. z
=1
Figure 1.28 shows all 3
cases
Figure
1.28.
by
Tetyana Butler
