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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
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