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Simple Contour Integration Question

  1. Dec 18, 2007 #1
    1. The problem statement, all variables and given/known data

    We want to find the value of the integral [tex]\int_{-\infty}^{\infty} \frac{e^{i2\pi ft}}{1+if}df[/tex] using contour integration.

    2. Relevant equations

    This should be solved by setting z = i*f and using the residue theorem. The equation has a pole at z = -1.

    3. The attempt at a solution

    I already have the solution, but I have trouble understanding it, so I hope someone here can enlighten me.

    We divide the contour integral into two cases: t>0 and t<0. The diagram is shown here.

    I know the key here is to draw the semi-circle so that the integral along the semi-circle disappears as we extend the limits to infinity, but I'm unclear on how our solution came up with the contours. Especially the second one, since I was under the impression that all contours had to be "counterclockwise" - or at least, if we were walking along the line, the area enclosed in our contour should be on our left. This doesn't appear to be the case for the second case.

    Can anyone help? My main problem is understanding how to choose where the semi-circles should go, but any clarification on the direction of the contour integral in the second case would be appreciated as well.

    Thanks!
     
  2. jcsd
  3. Dec 19, 2007 #2

    Hootenanny

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    I don't see why you have to use two contours in this case. I find the simplest way to compute these integrals would be to take a semi-circular contour with it's diameter being the real axis and an indented contour around the singularity on the boundary.

    However, to answer your question, if we have a contour C oriented clockwise then;

    [tex]\int_{-C}f(t)dt = -\int_{C} f(t)dt [/tex]
     
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