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Math Help - complex analysis: integrals

  1. #1
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    complex analysis: integrals



    how can i do this, applying Rouche Theorem and Cauchy Argument Principle?
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  2. #2
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    Use Cauchy's Argument Principle.

    1) \frac{1}{2\pi i}\oint_{\Gamma}\frac{f'(z)}{f(z)}dz = \mathbb{Z} - \mathbb{P} where \mathbb{P} is the number of poles and \mathbb{Z} is the number of zeros (both counting multiplicity). Thus, this equation tells us that \mathbb{Z} - \mathbb{P} = 2 but \mathbb{P} = 0 because f(z) ia analytic on \Omega. Thus, the function f(z) has two zeros z_1 and z_2 within the unit disk (and I let z_1 = z_2 so if it is the same zero of multiplicity two).

    2) \frac{1}{2\pi i}\oint_{\Gamma}z\frac{f'(z)}{f(z)}dz = z_1 + z_2. Thus, z_1+z_2 = 0.

    3) \frac{1}{2\pi i}\oint_{\Gamma}z^2 \frac{f'(z)}{f(z)}dz = z_1^2+z_2^2 = \frac{1}{2}.

    Now solve for z_1 \mbox{ and }z_2.
    Last edited by ThePerfectHacker; December 4th 2007 at 07:52 AM.
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