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Math Help - Determine the number of zeros of a given function by using Rouche's Th

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    Determine the number of zeros of a given function by using Rouche's Th

    please help me to determine the number of zeros of the following function in the disk D(0,2) by using Rouche's Th:

    f(z)=z^2e^z-z, z\in D(0,2)

    z is a complex number!~
    Last edited by frankmelody; November 10th 2008 at 04:37 PM.
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    Quote Originally Posted by frankmelody View Post
    please help me to determine the number of zeros of the following function in the disk D(0,2) by using Rouche's Th:

    f(z)=z^2e^z-z, z\in D(0,2)
    Is there a problem with saying that z^2e^z\geq{z} and z^2e^z has two zeros on that range, thus the actual function has two? Or am I remembering the theorem wrong...
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    Quote Originally Posted by Mathstud28 View Post
    Is there a problem with saying that z^2e^z\geq{z} and z^2e^z has two zeros on that range, thus the actual function has two? Or am I remembering the theorem wrong...
    what do you mean by z^2e^z\geq{z}? I think z is a complex number here......
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    Quote Originally Posted by frankmelody View Post
    please help me to determine the number of zeros of the following function in the disk D(0,2) by using Rouche's Th:

    f(z)=z^2e^z-z, z\in D(0,2)

    z is a complex number!~
    Let g(z) = z^2e^z and h(z) = -z.
    Then on |z|=2 we have |g(z)| > |h(z)|.

    Therefore, g(z)+h(z) has as many zeros as g(z) in D(0,2)
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    Quote Originally Posted by ThePerfectHacker View Post
    Let g(z) = z^2e^z and h(z) = -z.
    Then on |z|=2 we have |g(z)| > |h(z)|.

    Therefore, g(z)+h(z) has as many zeros as g(z) in D(0,2)
    I am afraid you are wrong, you said on " |z|=2 we have |g(z)| > |h(z)|", but if you let z=-2, obviously you are wrong.
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