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Math Help - Rouche's Theorem

  1. #1
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    Rouche's Theorem

    Find the number of zeros of the following polynomial lying inside the unit circle;
    f(z)=z^9 - 2z^6 + z^2 - 8z -2


    I tried to use Rouche's Theorem
    for differentiable f and g and all points inside s
    if |f(z)-g(z)|<|f(z)| then
    f and g have same zeros in s.
    But which of the g(z) functions should I choose? -2z^6 or z^2 or -8z
    how can ı determine this?
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  2. #2
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    As far as i can remember you have to choose the dominating function within the region, i.e the one which grows the fastest.

    Within the unit circle, the dominating part of your function is -8z, so try choosing that and see if that works, i may be wrong though.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by AcCeylan View Post
    But which of the g(z) functions should I choose? -2z^6 or z^2 or -8z how can ı determine this?

    If |z|=1 then,

    |z^9 - 2z^6 + z^2  -2 |\leq |z^9|+|-2z^6|+|z^2|+|-2|=1+2+1+2=6<8=|-8z|
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