Polar form
Let
, where
and
 .
The precise meaning of the last equation, namely of
is that there exist exactly
distinct
roots for any non zero complex number
. You can check this by the following way: let
increase from 0 to
, you have distinct numbers. From
and further on, you get the same complex numbers again (you only added a
multiple of
to the argument...).
Example 1.5.2
Find the cubic roots of
. Thus:
The cubic roots of
are:
Example 1.5.3
Find the cubic roots of
.
. Thus:
The cubic roots of
are:
and
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![$\displaystyle =\sqrt[6]{2} \left( \cos \frac {\pi}{4} + i \sin \frac {\pi}{4} \...
...eft( \frac {\sqrt{2}}{2} + i \frac {\sqrt{2}}{2} \right) =2^{-1/3} + i 2^{-1/3}$](/images/maths/complex-analysis/polar-f/img260.png)
![$\displaystyle = \sqrt[6]{2} \left( \cos \frac {11 \pi}{12} + i \sin \frac {11 \pi}{12} \right)$](/images/maths/complex-analysis/polar-f/img262.png)
![$\displaystyle = \sqrt[6]{2} \left( \cos \frac {19 \pi}{12} + i \sin \frac {19 \pi}{12} \right)$](/images/maths/complex-analysis/polar-f/img264.png)
