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Contour integral involving gamma function

  1. Sep 30, 2014 #1

    CAF123

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    1. The problem statement, all variables and given/known data
    Evaluate the integral by closing a contour in the complex plane $$\int_{-\infty}^{\infty} dx e^{iax^2/2}$$

    2. Relevant equations
    Residue theorem


    3. The attempt at a solution
    My initial choice of contour was a semicircle of radius R and a line segment from -R to R. In the limit R to infinity, I would hopefully recover the integral in OP. But then I realized this was going to work because the integral over the semicircle would not vanish when R tended to infinity. My question is, what is the best way to see what the best contour should be for a given problem and how to determine the one right for this problem?

    Many thanks
     
  2. jcsd
  3. Sep 30, 2014 #2

    haruspex

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    By symmetry, you could conside only a semi infinite integral.
    Because of the exponential term, I suggest a rectangle will work better than an arc.
     
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