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Math Help - Integral on the Complex Plane

  1. #1
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    Integral on the Complex Plane

    At the end of my complex analysis class, there is an integral which I can't work out. The main problem I have is the appearance of a singularity on my contours. Does anyone know how to solve it. Unfortunately I've never learnt latex so here it is in word format.

    I = -∞Int∞ (cos(x) - 1)/(x^2) dx


    Thanks for any help or anyone who can convert that to latex.
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  2. #2
    A Plied Mathematician
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    Your integral can be written as follows:

    \displaystyle{\int_{-\infty}^{\infty}\frac{\cos(x)-1}{x^{2}}\,dx.}

    You can double-click the integral to see the LaTeX code that generated it.

    I agree that you can't integrate directly over a pole. What you're going to have to do is skirt the pole by doing an \epsilon semicircle around the pole. It's a first-order pole, by the way, since the numerator has a first-order zero at zero. Once you've computed the integral over the \epsilon semicircle, you can add to that the other two integrals from (-\infty,-\epsilon) and (\epsilon,\infty). Make sense?
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