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Topic: How to get this gaussian integral result?
Replies: 1   Last Post: Apr 4, 2012 4:29 AM

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 Scott Hemphill Posts: 196 Registered: 12/13/04
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 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@alumni.caltech.edu
"This isn't flying. This is falling, with style." -- Buzz Lightyear

Date Subject Author
4/3/12 simplerbysimpler@gmail.com
4/4/12 Scott Hemphill