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Complex numbers, basic problem

  1. Aug 17, 2010 #1


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    1. The problem statement, all variables and given/known data

    Know: modulus(z) < 3
    WTS: |Im(z2 - zbar + 6)| <12

    where zbar is the complex conjugate

    2. Relevant equations

    z = x + iy

    3. The attempt at a solution

    |Im(z2 - zbar + 6)|
    = |Im(x2 + 2i*x*y - y2 - x + iy + 6)|
    = |2xy + y|

    So I want to show |2xy + y|< 12

    I already proved it using maximization and Lagrange multipliers, but it seems like overkill, and I think there is some kind of arithmetic trick I am missing. Anyone see it?

    Last edited: Aug 17, 2010
  2. jcsd
  3. Aug 17, 2010 #2
    |Im(z)|[tex]\leq[/tex] |z|
    Then use the triangle inequality to derive an upper bound.
  4. Aug 17, 2010 #3


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    Homework Helper

    One thing to be careful about is to use:
    |z_{1}-z_{2}|\leqslant ||z_{1}|-|z_{2}||
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