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Math Help - Complex Analysis

  1. #1
    Senior Member Dinkydoe's Avatar
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    Complex Analysis

    Hee I have not yet a good idea how to start on this one.

    Let f(z)=\frac{p(z)}{q(z)} a rational function with \text{deg}(q)\geq \text{deg}(p)+2.

    Show that the sum of the residues of f is zero.

    Some guiding hints will be very much appreciated.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by Dinkydoe View Post
    Hee I have not yet a good idea how to start on this one.

    Let f(z)=\frac{p(z)}{q(z)} a rational function with \text{deg}(q)\geq \text{deg}(p)+2.

    Show that the sum of the residues of f is zero.

    Some guiding hints will be very much appreciated.
    Integrate f round a circle centred at the origin, with really large radius R. On the circle, |f(z)| < \text{const.}/R^2, so the absolute value of the integral will be of order 1/R. As R goes to infinity, the contour contains all the singularities, and the integral ( = 2\pi i\times(sum of residues)) goes to 0.
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  3. #3
    Senior Member Dinkydoe's Avatar
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    Ah thanks,that explains why the condition deg(p)<=deg(q)+2 is necessary
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