Find all $z$ where $ z^{6} = i \overline{z} $

Question

$ z^{6} = i*\overline{z} $

I attempted the following:

$r^6 * e^{6ti} = e^{i\pi/2} * re^{-ti}$

we divide and we get

$r^5 e^{7ti} = e^{i\pi/2}$

and then we see that

r = 1 and that $ t = \frac{\pi/2 + 2\pi k}{7}$

and from there we get $\pi /14, 5 \pi /14, 9 \pi/14, 13 \pi /14, 17 \pi/14, 3 \pi /2 $. But according to Wolfram Alpha there are more results.

What did I do wrong?

Answer 1

Any time you divide (also known as cancel), you should see whether the thing you are dividing by can be zero. If so, does that give another solution to your problem? If you do not do this check, your eventually computed solution set may not contain this solution.

In this case, you divided by $r \mathrm{e}^{-\mathrm{i}t}$, which can be zero when $r= 0$, which gives another solution, $z = 0$.

Also, your step size for $t$ is $\dfrac{2\pi}{7}$, so you will need to let $k$ range over $[0,6]$ or $[1,7]$ to get a full set of solutions before repeats.