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a bug in many variables integration ?

• To: mathgroup at smc.vnet.net
• Subject: [mg52467] a bug in many variables integration ?
• From: vivien lecomte <vivien.lecomte at th.u-psud.fr>
• Date: Sun, 28 Nov 2004 01:06:52 -0500 (EST)
• Reply-to: vivien.lecomte at th.u-psud.fr
• Sender: owner-wri-mathgroup at wolfram.com

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

• Follow-Ups:
  • Re: a bug in many variables integration ?
    • From: DrBob <drbob@bigfoot.com>