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Thread: Number of zeros in each domain (Complex Analysis)

  1. #1
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    Number of zeros in each domain (Complex Analysis)

    Find the number of zeros in each of the domains.

    (1) $\displaystyle z^{10} -10z^7 +2iz^4 -z^3 +i$

    (a) $\displaystyle {|z|<\frac{1}{2}}$

    (b) $\displaystyle {|z|<1}$

    (c) $\displaystyle {|z|<2}$

    (d) $\displaystyle {|z|<3}$

    I have no idea how to approach these. If someone could help me out or point me in the right direction for one of these, I would appreciate it. Thanks!
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  2. #2
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    Use Rouché's theorem. In each case, find the dominant term (or terms) for that value of |z|, and apply the theorem.

    To take part (b) as an example, when |z|=1 it's clear that $\displaystyle |z^{10} +2iz^4 -z^3 +i|<|-10z^7 |$. By Rouché's theorem, $\displaystyle z^{10} -10z^7 +2iz^4 -z^3 +i$ has the same number of zeros inside the circle |z|=1 as $\displaystyle -10z^7$, namely 7.
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    Very nice.
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