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Help needed on complex exponential equation

  1. Nov 18, 2012 #1

    I have come across this equation:

    zn-1=[itex]\bar{z}[/itex], z[itex]\in[/itex]ℂ* (ℂ*:=ℂ\0)

    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 :)
  2. jcsd
  3. Nov 18, 2012 #2

    Write [itex]\,z=re^{it}\,\,,\,\,0\leq t<2\pi\,[/itex] , so that

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

  4. Nov 18, 2012 #3
    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. :)
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