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)
• 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

```

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