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Math Help - Complex Numbers III

  1. #1
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    Question Complex Numbers III

    One final problem I have to solve is:

    If w=(z-i)/(z+i) and z lies below the real axis, show that w lies outside the unit circle |w|=1.
    How will w move as z travels along the real axis from -infinity to +infinity.

    Many thanks once more to those who are able to provide help or advice.
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  2. #2
    Senior Member JaneBennet's Avatar
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    Let z = x+yi, where x and y are real and y < 0. Substitute this into w, simplify, and show that |w| > 1.

    EDIT: Iíve just tried the problem, and I think a simpler method is to express z in terms of w and let w = u+vi instead. After substituting and simplifying, you should get

    \mathrm{Im}(z)\ =\ \frac{1-u^2-v^2}{(1-u)^2+v^2}

    Since Im(z) < 0, the rest is straightforward.

    EDIT again: For the second part of the question
    How will w move as z travels along the real axis from -infinity to +infinity.
    set Im(z) = 0.
    Last edited by JaneBennet; December 20th 2007 at 04:28 AM.
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  3. #3
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    Thanks to Jane Bennet for your help, however, I attempted the question and could not work out how the equation simplifies to your expression.

    Smiler
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  4. #4
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    Quote Originally Posted by smiler View Post
    could not work out how the equation simplifies to your expression
    This may not help but:
    w = \frac{{z - i}}{{z + i}}\left( {\frac{{\overline z  - i}}{{\overline {z + i} }}} \right) = \frac{{z\overline z  - zi - \overline z i - 1}}{{\left| {z + i} \right|^2 }} = \frac{{x^2  + y^2  - 2xi - 1}}{{x^2  + \left( {y + 1} \right)^2 }} where z = x + yi.
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