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>

```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

```

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