How to Solve the Complex Equation zn-1=\bar{z}?

[freddyfish]

Hi

I have come across this equation:

z^n-1=$\bar{z}$, z$\in$ℂ^* (ℂ^*:=ℂ\0_ℂ)

There are numerous obvious equalities that can be used, but I don't seem to reach a satisfying final answer.

Any help would be appriciated.

Thanks in advance :)

 

[DonAntonio]

Hi

I have come across this equation:

z^n-1=$\bar{z}$, z$\in$ℂ^* (ℂ^*:=ℂ\0_ℂ)

There are numerous obvious equalities that can be used, but I don't seem to reach a satisfying final answer.

Any help would be appriciated.

Thanks in advance :)

Write $\,z=re^{it}\,\,,\,\,0\leq t<2\pi\,$ , so that

$$z^{n-1}=\bar z\Longleftrightarrow r^{n-1}e^{(n-1)it}=re^{-it}\Longrightarrow r=1\,\,,\,(n-1)t=-t\pmod {2\pi}\Longleftrightarrow nt=0\pmod{2\pi}$$

DonAntonio

 

[freddyfish]

That's what I got too, using a somewhat different method, but I felt that my answer didn't include all the solutions to this equation for some reason. But I suppose my gut feeling betrayed me on this one.

Thank you for your time and effort. :)