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Evaluate Fourier Integral

  1. Dec 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that

    integral from 0 - > infinity (cos(alpha*x)/(alpha^2 + 1))dalpha = (pi/2)exp(-x)

    2. Relevant equations



    3. The attempt at a solution

    cos(alpha*x) = (1/2)(exp(i*alpha*x)+exp(-i*alpha*x))

    Really don't know where to go from here.
     
  2. jcsd
  3. Dec 9, 2012 #2

    Dick

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    Use contour integrals and the residue theorem. A lot of Fourier integrals need to be done that way.
     
  4. Dec 9, 2012 #3
    The problem says to evaluate directly with the Fourier Integral Theorem.
     
  5. Dec 9, 2012 #4

    Dick

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    Ok, so what is the Fourier Integral Theorem?
     
  6. Dec 9, 2012 #5
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