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Math Help - Application of Rouche's theorem

  1. #1
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    Application of Rouche's theorem

    I want to estimate the number of zeros for the function, by using Rouche's theorem.

    f(z)=z^2e^z-z in the region D(0,2), which is a disc around 0 with radius 2.

    I can only find and perform easy examples for polynomials, this one has been bugging me for a while now. No clue in what direction I have to look to pick a function to compare f(z) with.
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  2. #2
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    Hmm. I would break up your domain into two pieces, depending on whether the imaginary part of z is negative or non-negative. One obvious root is the origin. In the right half-plane intersected with your domain, the exponential function makes the z^{2} larger, does it not? And smaller in the left-half plane intersected with your domain, correct? Just an idea or two.
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  3. #3
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    Because LaTeX isn't fully working yet, the function mentioned is f(z) = z^2 * e^z - z on the open disc around 0 with radius 2.

    @Ackbeet, thanks for the reply but I want to make use of Rouche's theorem.
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  4. #4
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    brouwer, I was trying to help you use Rouche's theorem. As stated in Gamelin, p. 229, Rouche's theorem runs like this:

    Let D be a bounded domain with piecewise smooth boundary D'. Let f(z) and h(z) be analytic on D union D'. If |h(z)| < |f(z)| for z in D', then f(z) and f(z) + h(z) have the same number of zeros in D, counting multiplicities.

    I was trying to help you find function h(z). I was suggesting that you break up your domain into two pieces and apply Rouche's theorem on each piece.
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