A curious integral and a trigonometric identity
- To: mathgroup at smc.vnet.net
- Subject: [mg111969] A curious integral and a trigonometric identity
- From: David Papp <dpapp at rutcor.rutgers.edu>
- Date: Sun, 22 Aug 2010 08:11:13 -0400 (EDT)
I have recently stumbled upon this:
In[1] := Integrate[1/((1 + x^2)*(1 + y^2)), {x, -Infinity, Infinity},
{y, x, Infinity}]
Out[1] := 0
I saw that a number of bugs with symbolic multiple integrals have been
reported before, but this seems to be a new one. I get the same wrong
answer not only with the 7.0.1 version, but also with 6.0 and 5.1.
This might be related to the following "identities":
In[2] := FullSimplify[Integrate[1/(1 + y^2), {y, x, Infinity},
Assumptions -> (x > 0)]]
Out[2] := ArcCot[x]
In[3] := FullSimplify[Integrate[1/(1 + y^2), {y, x, Infinity},
Assumptions -> (x < 0)]]
Out[3] := 1/2 (\[Pi] - 2 ArcTan[x])
In[4] := Reduce[ArcCot[x] == 1/2 (\[Pi] - 2 ArcTan[x]), x]
Out[4] := True
The last one is disturbing because Reduce is supposed to give exact (not
generic) answers. I get this only in v7. (The earlier versions cannot
simplify it: "This system cannot be solved with the methods available to
Reduce".)
If these are indeed bugs, I am not sure where to send it within Wolfram.
Any suggestions? Thank you,
David