I know that the roots of a complex number can be calculated by $\sqrt[n]{z}$ =$\sqrt[n]{|z|}$ * $e^i\frac{\theta+2k\pi}{n}$ and i suppose that using $\sqrt[n]{z}$ = $e ^\frac {log(z)}{n}$ should also give the same result. The problem i found is that when using log i can only find one of the acceptable results. Can someone explain me why?
Note: My guess is because Ι am limiting my $\arg(z)$ to $[0,2\pi]$ when using log
Thanks guys
$\log(z)$ is not a single valued function. Because for any fixed $z$ there are many $w$ such that $e^w = z$, in particular $e^{w+i2\pi k}$ will work for any integer $k$. Therefore $\log(z) = w+i2\pi k$ for any integer $k$. That is, you get infinitely many values. So yes, it is because you are limiting your arg to $[0,2\pi]$, you are only getting one answer for the root. Try using argument in the range $[2\pi,4\pi]$ and you'll find you get the second possible root. If you used the range $[4\pi,6\pi]$ you will get the third possible root. And so on. If you use all the values of $\log(z)$ you will get all the roots.