Equation in the complex plane 8 z = i | z | 3 z ¯ ?

Question

Equation in the complex plane   8  z  =  i      |    z            |        3              z      ¯      ?

Austin Solis · Accepted Answer

Let   r  =      |    z      |   and   θ  =  arg  ⁡  (  z  ) so that   z  =  r      e          i      θ      . Then       z    ¯    =  r      e          −      i      θ       and so the equation becomes  8  r      e          i      θ        =  i      r    4        e          −      i      θ        .Note that   i  =      e          i                        π          2                    , so the right-hand side equals       r    4        e          i      (                        π          2                    −      θ      )      . Rearranging terms the above becomes  r      e          i      θ        ⋅      (    8    −          r      3              e              i        (                              π            2                          −        2        θ        )              )    =  0  ,which shows that either r=0 or  8  −      r    3        e          i      (                        π          2                    −      2      θ      )        =  0  ,in which case r=2 and             π      2        −  2  θ  =  2  k  π for some   k  ∈      Z  . Then   θ  =  (            1      4        −  k  )  π, so   θ  =            1      4        π or   θ  =            5      4        π as   θ  ∈  [  0  ,  2  π  ). This shows that there are precisely three solutions, which are  z  =  ±      2    (  1  +  i  )     or     z  =  0.

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