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Long story short, I need to compute an autocorrelation integral. Here's the problem:

There are two arbitrary gaussian pulses, one following the other by a fixed distance. By computing the autocorrelation over space(not time) and taking the derivative of the space-shift autocorrelation and setting it equal to zero, important information hopefully could be obtained.

The mathematics of this would be as followed:

[itex]\frac{\partial}{\partial\tau}\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(Ae^{-a(x-c-\tau)^2}+Be^{-b(x-d-\tau)^2})dx=0[/itex]

[itex]\int_{-\infty}^{\infty}(Ae^{-a(x-c)^2}+Be^{-b(x-d)^2})(a(x-c-\tau)Ae^{-a(x-c-\tau)^2}+b(x-d-\tau)Be^{-b(x-d-\tau)^2})dx=0[/itex]

I am NOT asking anyone to do this for me - I'll do it myself but I just need some ideas or directions on how to go about it.

I have experience in fourier transforms, complex analysis and calculus of course. I've considered doing a complex contour integral but I'm not sure how reasonable that is after seeing how big of a pain the normal gaussian contour integral is. I've considered fourier transforms a little - I didn't immediately see much help due to the fourier transform of a gaussian just being another gaussian. I've thought about parametrization or even centering the integral about the center of the two gaussians but I don't know where to start I guess.

It's clearly a bound integral but is it just too impossibly hard to try?

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# Nasty Autocorrelation Integral

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