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Re: a bug in many variables integration ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg52482] Re: [mg52467] a bug in many variables integration ?
*From*: DrBob <drbob at bigfoot.com>
*Date*: Mon, 29 Nov 2004 01:22:29 -0500 (EST)
*References*: <200411280606.BAA06640@smc.vnet.net>
*Reply-to*: drbob at bigfoot.com
*Sender*: owner-wri-mathgroup at wolfram.com
Here's an answer from version 5.1:
q[u_, v_] := (2 - c)*u^2 - 2*e*u*v + c*v^2
Integrate[Exp[-q[u, v]],
{u, -Infinity, Infinity},
{v, -Infinity, Infinity}]
(1/Sqrt[c])*(Sqrt[Pi]*
If[Re[c + e^2/c] < 2,
Sqrt[Pi]/Sqrt[2 - c -
e^2/c], Integrate[
E^((((-2 + c)*c + e^2)*u^2)/
c), {u, -Infinity,
Infinity}, Assumptions ->
Re[c + e^2/c] >= 2]])
Bobby
On Sun, 28 Nov 2004 01:06:52 -0500 (EST), vivien lecomte <vivien.lecomte at th.u-psud.fr> wrote:
> Hi all,
>
> Mathematica 5 behaves in a strange way when integrating
> gaussian functions. Consider the following quadratic form
> in u and v :
>
> q[u_, v_] := (2 - c) u^2 - 2 e u v + c v^2
>
> Mathematica 5.0 gives 0 when computing its interal over
> R^2 :
>
> In[2]:= Integrate[Exp[-q[u, v]], {u, -Infinity, Infinity},
> {v, -Infinity, Infinity}]
>
> Out[2]= 0
>
> without any warning. This is of course absurd, as :
>
> Exp[-q[u, v]] > 0 ...
>
> q can be written as {{u, v}}.Q.{{u}, {v}} where M is the
> following matrix :
>
> Q := {{ 2-c , -e },
> { -e , c }}
>
> When |e|<1, one can easily see that there always exists
> values of c such that Q is a positive definite matrix,
> prooving that the integral above exists, and is :
>
> Pi / Sqrt[ Det[Q] ]
>
>
> This is the result given by Mathematica 4.1 :
>
>
> Pi
> Out[2]= ------------------------
> 2
> e
> Sqrt[c] Sqrt[2 - c - --]
> c
>
> So, where is the bug, if any ?
> - Does math5 forget to tell me he makes contradictory assumptions
> on the parameters c and e ?
> - Does he proceeds the 2-variable integration in some wrong order ?
>
>
> With best wishes,
>
> Vivien Lecomte
>
>
>
>
>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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