How to Compute an Autocorrelation Integral for Gaussian Pulses?

[krobben92]

Hi guys,

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:
\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
\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

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?

 

[mfb]

You can reduce the problem to several integrals of the types $\int dx e^{-(x+d)^2 - (x-d)^2}$, $\int dx x e^{-(x+d)^2 - (x-d)^2}$ and maybe something I missed and look for solution methods for those integrals.

 

[krobben92]

Yes, that's definitely one way to do it. However, this may take a couple dozen sheets of paper and a few hours considering the factoring. I guess after seeing so many tricks in math classes I just assumed there might be a quick way around this... But real world problems versus classroom problems aren't a fair comparison I suppose.

 

[krobben92]

Well I tried it the old fashion way and it turned out better than I thought - but I don't think \tau can be solved for explicitly.

\frac{-A^{2}}{2\sqrt{2a}}e^{\frac{-a\tau^{2}}{2}}+\frac{-B^{2}}{2\sqrt{2b}}e^{\frac{-b\tau^{2}}{2}} = \frac{2ABe^{\frac{-ab(j(j-2k)+k^{2})}{a+b}}}{(a+b)\sqrt{a+b}}e^{\frac{-ab\tau^{2}}{a+b}}(\tau cosh(\frac{2ab\tau(j-k)}{a+b})-(j-k)sinh(\frac{2ab\tau(j-k)}{a+b}))

Any other ideas? I'm not an expert on Fourier transforms but I'm beginning to think that's the only way because it should be a multi-valued answer - just not sure how to approach it.