Re: How to get this gaussian integral result?

*To*: mathgroup at smc.vnet.net*Subject*: [mg125820] Re: How to get this gaussian integral result?*From*: Scott Hemphill <hemphill at hemphills.net>*Date*: Wed, 4 Apr 2012 04:28:08 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jledpe$h35$1@smc.vnet.net>*Reply-to*: hemphill at alumni.caltech.edu

simplerbysimpler at 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 closed-form 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 positive-definite. Scott -- Scott Hemphill hemphill at alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear