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Math Help - The Zeros of an Analytic Function

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    The Zeros of an Analytic Function

    Determine how many zeros of the given function lie in the given annulus.

    (1) z^3-3z+1 in  1<|z|<2

    (2) z^4 -2z -2 in  \frac{1}{2} <|z|< \frac{3}{2}

    Thanks for any help!
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  2. #2
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    Quote Originally Posted by shadow_2145 View Post
    Determine how many zeros of the given function lie in the given annulus.

    (1) z^3-3z+1 in  1<|z|<2
    Intermediate value theorem: the function has three real zeros, in the intervals (-2,-1), (0,1), (1,2).

    Quote Originally Posted by shadow_2145 View Post
    (2) z^4 -2z -2 in  \frac{1}{2} <|z|< \frac{3}{2}
    Rouché's theorem: |2(z+1)|<|z^4| on |z| = 3/2; |z^4-2z|<2 on |z| = 1/2.
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