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Contour integration to find real integral

  1. Sep 5, 2008 #1
    [tex]g(z) = \frac{e^{imz}}{z^2}[/tex]

    m is real, m > 0

    Integrate g(z) around a suitable closed contour to find the principle value of

    [tex]\int^{\infty}_{-\infty}\frac{e^{imx}}{x^2}dx[/tex]


    Obviously parts of the contour must lie on the negative and positive real axes. There's a double pole at z=0 so we need to indent around it. My first instinct is to integrate anti-clockwise around a large semi-circle C_R in the upper half plane, from -R to -r on the negative real axis, clockwise around a small semi-circle C_r, and from r to R. In the limit r to 0, R to infinity, the desired integral appears as part of the contour integration.
    But the integration around C_r seems to diverge as r goes to zero. I can't think of any other way to approach this problem, can anyone help?
     
  2. jcsd
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