1. ## Complex analysis

Hi, need some help to clear some doubts, have test on monday

Find all the values of :
$\displaystyle (1+\sqrt3i)^{1/3}$

I got : $\displaystyle z=2^{1/3}*e^{i({2k\pi}+\pi/3})$ ,k = 0,1,2

$\displaystyle z_{0}=2^{1/3}*e^{i\pi/9}$, $\displaystyle z_{1}=2^{1/3}*e^{i7\pi/9}$,$\displaystyle z_{2}=2^{1/3}*e^{-i5\pi/9}$

Can somebody chek my answers, coz am not sure they are good, btw i see in my notes that the nth roots of unity forms a regular polygon, in our case it should form an equilateral triangle ? or is it only when $\displaystyle z^x=1$ that it holds ??

2. Yea, that's right. The central issue is $\displaystyle \frac{\Theta}{n}+\frac{2k\pi}{n}$ in the expression $\displaystyle z^{1/n}=r^{1/n}e^{i/n(\Theta+2k\pi)}$.

So you're breaking up the circle into $\displaystyle \frac{2k\pi}{n}$ sectors of equal size starting at the point $\displaystyle \Theta/n$ with a root at each terminal point of each sector so of course they're going to be equally spaced around a circle of radius $\displaystyle Abs(z)^{1/n}$ and therefore at the points of a regular polygon.

Here's an idea. Write a Mathematica application which illustrates the roots of $\displaystyle z^{1/n}$ as red points in the complex plane for a variable sized z for n running from 2 to 10. Sort of like this:

Code:
Manipulate[
pts = N[w /. Solve[r*Exp[I*tval] == w^n,
w]]; points =
(Point[{Re[#1], Im[#1]}] & ) /@ pts;
Show[Graphics[{Red, PointSize[0.02],
points}], PlotRange ->
{{-3, 3}, {-3, 3}}, Axes -> True],
{r, 1, 5}, {tval, 0, Pi/2, 0.1},
{n, 2, 10, 1}]