Evaluate Int: Gaussian Wavepacket & Fourier Transform

[thegaussian]

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!

 

[tim_lou]

resolve cos into exponentials and complete the square. Do a contour shift (or just pretend i is just a parameter)

 

[ice109]

resolve cos into exponentials and complete the square. Do a contour shift (or just pretend i is just a parameter)

what does that mean?

 

[tim_lou]

what does that mean?

well, I believe that if you work out the integral, you'll get something like
$$\int_{-\infty}^{\infty} e^{-(a+ib)(x-(c+id))^2} dx =\int_C e^{-(a+ib)z^2} dz$$

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.