
Re: How to get this gaussian integral result?
Posted:
Apr 4, 2012 4:29 AM


simplerbysimpler@gmail.com writes:
> Integrate[ > x1^(n1) x2^(n2) x3^(n3) Exp[ > a x1 x1 + b x2 x2 + c x3 x3 + d12 x1 x2 + d23 x2 x3 + d13 x1 x3 + > d], {x1, Infinity, Infinity}, {x2, Infinity, > Infinity}, {x3, Infinity, Infinity}]. n1,n2,n3 are Natural numbers. I still cannot get the general result ,although mathematica takes so much time. Can you help me about this?
I think the basic problem is that this integral doesn't converge for most of the values of a, b, c, d12, d23 and d13. If I wanted the value when it converges, I would change variables in such a way that the "Exp" expression reads "Exp[ u1 u1  u2 u2  u3 u3]". You then end up with a lot of terms, but each of them has a closedform solution.
The key to the variable change is an eigensystem analysis of the matrix
{{ a, d12/2, d13/2 }, { d12/2, b, d23/2 }, { d13/2, d23/2, c }}
and the integral converges whenever this matrix is positivedefinite.
Scott  Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style."  Buzz Lightyear

