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Complex
Analysis. FreeTutorial
14.1
n-th root of complex numbers
14.1.1
Roots of unity
14.1 n-th
root of complex numbers
To find the n-th
root of a complex number w
0 we have to solve
the equation
zn
= w.
(1.26)
Let w
= r(cos +i
sin),
z
=  (cos +i
sin).
Then (1.26) takes the form
n(cos
n+i
sin
n)
= r(cos+i
sin).
(1.27)
The equation (1.27) is fulfilled if
n
= r
and n=
and we obtain the root
z
=  ,
(1.28)
where 
is the positive
n-th root of
the positive number r.
(1.28) is not the only solution of (1.27).
It is also fullfield for all angles, satisfied
 ,
where k
is any integer.
However, only the values k
= 0, 1,
…, n
- 1 give different
values of z.
The complete solution of the (1.27) is
given by
z
=  ,
k
= 0, 1,
…, n
- 1.
(1.29)
Every non zero complex number has exactly
n
distinct n
th roots.
It means that every number
has two square roots,
three cube roots, four
fourth roots, ninety
ninetieth roots, and so on.
They have the same modulus and their arguments
differ by 
, k
= 0, 1,
…, n
- 1.
Geometrically, the n
th roots are the vertices of a regular
polygon with n
sides.
To find n
th roots it is a four-step process.
1. Plot your number r(cos+i
sin),
that you want to take the root of.
2. Plot a circle of radius .
It is the length of all roots.
3. The first root has direction /n.
4. The other n
roots are equally spaced on the circle
of radius .
Top
14.1.1
Roots of unity
The roots of equation
zn
= 1
(1.30)
are called n-th
roots of unity.
The solution of the (1.30) is given by
z
= cos( )
+ i
sin(),
k
= 0, 1,
…, n
- 1.
(1.31)
We remember that every non
zero complex number has exactly n
distinct n
th roots. It means that unity has two
square roots, three cube
roots, four fourth roots,
and so on.
They have the same modulus
= 1
and their arguments differ by
, k
= 0, 1,
…, n
- 1.
The n-th
roots of unity are located on the unit
circle of the complex plane. They form
the vertices of a n-sided regular polygon
with one vertex on 1.
Figure 1.29 shows all distinct roots of
unity zn
= 1
for n
= 1, 2, ..., 14.
Figure
1.29. Roots of unity
There is one first root
of unity, equal to 1.
The square roots of unity (n
= 2) are 1
and -1.
The third (cubic) roots of unity are 1,
 .
The fourth roots of unity are 1,
i,
-1, -i.
by
Tetyana Butler
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