Polar form.

Let $u=r ( \cos \theta + i \sin \theta)$ , where $r>0$ and $\theta \in \mathbb{R}$ .

$$z^n=u \Leftrightarrow \begin{cases}\vert z^n\vert=r \ \operatorn... ...eratorname{arg}\nolimits (z)= \frac {\theta}{n} + \frac {2k \pi}{n} \end{cases}$$

The precise meaning of the last equation, namely of

$$\operatorname{arg}\nolimits (z)= \frac {\theta}{n} + \frac {2k \pi}{n}$$

is that there exist exactly $n$ distinct $n^{th}$ roots for any non zero complex number $u$ . You can check this by the following way: let $k$ increase from 0 to $n-1$ , you have distinct numbers. From $k-n$ and further on, you get the same complex numbers again (you only added a multiple of $2 \pi$ to the argument...).

Example 1.5.2   Find the cubic roots of $u=-1+i$ .

$u=1-i = \sqrt{2} \left( \cos \frac {3 \pi}{4} = i \sin \frac {3 \pi}{4} \right)$ . Thus:

$$z^3=-1+i \Leftrightarrow \begin{cases}\vert z\vert^3= \sqrt{2} \\... ...\operatorname{arg}\nolimits (z) =\frac {\pi}{4} + \frac {2k \pi}{3} \end{cases}$$

The cubic roots of $u$ are:

$$z_1 =\sqrt[6]{2} \left( \cos \frac {\pi}{4} + i \sin \frac {\pi}{4} \... ...eft( \frac {\sqrt{2}}{2} + i \frac {\sqrt{2}}{2} \right) =2^{-1/3} + i 2^{-1/3}$$

$$z_2 = \sqrt[6]{2} \left( \cos \frac {11 \pi}{12} + i \sin \frac {11 \pi}{12} \right)$$

$$z_3 = \sqrt[6]{2} \left( \cos \frac {19 \pi}{12} + i \sin \frac {19 \pi}{12} \right)$$

Example 1.5.3   Find the cubic roots of $u=-2+2i \sqrt{3}$ .

$u=-1+i \sqrt{3}= 8 \left( \cos \frac {2 \pi}{3} = i \sin \frac {2 \pi}{3} \right)$ . Thus:

$$z^3=-1+i \sqrt{3} \Leftrightarrow \begin{cases}\vert z\vert^3= 8 ... ...peratorname{arg}\nolimits (z)= \frac {2 \pi}{9} + \frac {2k \pi}{3} \end{cases}$$

The cubic roots of $u$ are:

$$2 \left( \cos \frac {2 \pi}{9} + i \sin \frac {2 \pi}{9} \right), \; 2 \left( \cos \frac {8 \pi}{9} + i \sin \frac {8 \pi}{9} \right) \; \; 2 \left( \cos \frac {14 \pi}{9} + i \sin \frac {14 \pi}{9} \right).$$