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Math Help - Residue theorem

  1. #1
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    Residue theorem

    Hey guys.
    So I got this integral I need to solve, of curse using the residue theorem.
    The thing is, that I don't understand the curve.
    I know that whenever Z^2 = integer, this function has a singularity point because e^(2*pi*i*n) = 1.
    But again, I'm not sure what this curve has enclosed in.

    Thanks.
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  2. #2
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    Quote Originally Posted by asi123 View Post
    Hey guys.
    So I got this integral I need to solve, of curse using the residue theorem.
    The thing is, that I don't understand the curve.
    I know that whenever Z^2 = integer, this function has a singularity point because e^(2*pi*i*n) = 1.
    But again, I'm not sure what this curve has enclosed in.

    Thanks.
    The poles occur when e^{2\pi iz^2} - 1 = 0 \implies 2\pi i z^2 = 2\pi i k, k\in \mathbb{Z}.
    Therefore, the poles of this function are at z^2 = k.

    We are also told that n< R^2 < n+1 for some integer n\geq 0.
    For the point of illustration say n=3 then 3<R^2 < 4.
    By definition \Gamma_R is the circle |z| = R.
    The poles need to occur inside this circle, so |z| < R \implies |z^2| < R^2.
    Thus, the poles are all k\in \mathbb{Z} with |k| < R^2.
    In this case k=0,\pm 1, \pm 2, \pm 3.

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