Complex
Analysis. FreeTutorial
12.1
De Moivre's theorem
12.1.1
De Moivre's theorem for fractional powers
12.1 De Moivre's theorem
It was showen (1.19) and
(1.20) that the product of two complex
numbers:
z1
= r1(cos+i
sin),
z2
= r2(cos+i
sin)
can be written in the form
z1z2
=r1r2[(coscos
 sinsin)
+
+ i(sincos
+ cossin)].
If z1
= z2
= z
= r(cos+i
sin),
the product
z1z2
= z^{2}
=
r^{2}(cos(2)+i
sin(2)).
(1.23)
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.
For r
= 1 we obtain De
Moivre’s formula:
(cos+i
sin)^{n}
= cos(n)+i
sin(n).
(1.25)
The formula connects complex numbers and
trigonometry.
It is easy to use this formula to find
explicit expressions for the nth
roots of unity.
Top
12.1.1
De Moivre's theorem for fractional powers
De Moivre's theorem is not
only true for the integers. It can be
extended to fractions.
Let n
= p/q
in (1.24). Then we have
z^{p/q}
= r^{p/q}{cos((p/q))+i
sin((p/q))}.
(1.26)
For r
= 1 we obtain De
Moivre’s formula for fractional powers:
(cos+i
sin)^{p/q}^{}
= cos((p/q))+i
sin((p/q)).
(1.27)
Example
1
Calculate (cos(/4)+i
sin(/4))^{1/3}.
By De Moivre's theorem for
fractional powers
(cos(/4)+i
sin(/4))^{1/3}
= cos(/12)+i
sin(/12).
It is the first cube root of cos(/4)+i
sin(/4).
By the Fundamental theorem
of algebra, the equation of degree n
has n
roots. A complex number has 3
cube roots.
cos(/12+2/3)+i
sin(/12+2/3)
is the second cube root of cos(/4)+i
sin(/4).
cos(/12+4/3)+i
sin(/12+4/3)
is the third cube root of cos(/4)+i
sin(/4).
by
Tetyana Butler
