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Integral from Hell?

  1. Oct 29, 2008 #1
    I'm doing a fourier transform of a gaussian wavepacket, so I can get the momentum representation of the wave... To progress I need to evaluate the following integral:

    Int{exp[-(sigma^2.x^2 + bx)/4k].cos[(tx^2 - cx)/8k]}dx

    with sigma, b,k,t and c all being constants, and the limits being ±infinity.
    Any help would be much appreciated!
     
  2. jcsd
  3. Oct 29, 2008 #2
    resolve cos into exponentials and complete the square. Do a contour shift (or just pretend i is just a parameter)
     
  4. Nov 3, 2008 #3
    what does that mean?
     
  5. Nov 3, 2008 #4
    well, I believe that if you work out the integral, you'll get something like
    [tex]\int_{-\infty}^{\infty} e^{-(a+ib)(x-(c+id))^2} dx =\int_C e^{-(a+ib)z^2} dz[/tex]

    where the contour for z is not the real line but shifted by some c+id. One may then argue that since there are no poles anywhere, we can change the contour back to the real line and get a standard gaussian integral. Of course, usually people (at least for me) just pretend i is a real parameter and crank the integral through.
     
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