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Math Help - complex number question

  1. #1
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    complex number question

    i can do first part of this question. i need help with the last part


    find roots of  z^3 - 1 = 0
    show if one complex root is w show the other is w^2
    show Re(\frac{1}{1+w^r}) = 1/2 for r=1,2,3

    i need help for next part

    if  z_1^2 +z_2^2+z_3^2- z_1 z_2-z_2 z_3-z_3 z_1 = 0

    show
    z_1 = -wz_2 -w^2 z_3 or z_1 = -wz_3 -w^2 z_2
    deduce three complex numbers are in a equaliteral triangle
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  2. #2
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    Quote Originally Posted by stud_02 View Post
    if  z_1^2 +z_2^2+z_3^2- z_1 z_2-z_2 z_3-z_3 z_1 = 0

    show
    z_1 = -wz_2 -w^2 z_3 or z_1 = -wz_3 -w^2 z_2
    One way (not very elegant) is to write the equation as a quadratic in z_1 and solve it:

    z_1^2 - (z_2+z_3)z_1 + (z_2^2+z_3^2 - z_2z_3) = 0,

    \begin{aligned}z_1 &= \tfrac12\bigl(z_2+z_3 \pm\sqrt{6z_2z_3-3z_2^2-3z_3^2}\bigr)\\[0.5ex] &= \tfrac12\bigl(z_2+z_3 \pm\sqrt3i(z_2-z_3)\bigr) \\[0.5ex] &= \tfrac12(1\pm\sqrt3i)z_2 +\tfrac12(1\mp\sqrt3i)z_3.\end{aligned}

    That gives the required result, since the complex cube roots of 1 are \tfrac12(-1 \pm\sqrt3i).
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