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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



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