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Math Help - Complex numbers question

  1. #1
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    Complex numbers question

    "If z= cis() and w = (1+z)/(1+z*) , show that w = cis "

    Help! I have tried, but failed on this one!! It is high school level maths!
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  2. #2
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    Quote Originally Posted by karldiesen View Post
    "If z= cis() and w = (1+z)/(1+z*) , show that w = cis "
    This is a complicated exercise in factoring.
    First write z=\cos(\phi)+i\sin(\phi).
    Then note that  \frac{{1 + z}}{{1 + \overline z }} = \frac{{\left( {1 + z} \right)^2 }}{{\left| {1 + z} \right|^2 }}.

    But \left| {1 + z} \right|^2  = \left( {1 + \cos (\phi )} \right)^2  + \sin ^2 (\phi ) = 2 + 2\cos (\phi ).

    Also \left( {1 + z} \right)^2  = 1 + 2\cos (\phi ) + 2i\sin (\phi ) + \cos (2\phi ) + i\sin (2\phi ) = \cos (\phi )\left[ {2 + 2\cos (\phi )} \right] + i\sin (\phi )\left[ {2 + 2\cos (\phi )} \right].

    I will let you have the fun of doing the details.
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