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MathGroup Archive 2004

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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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