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Re: Incorrect symbolic improper integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg103625] Re: Incorrect symbolic improper integral
  • From: Bill Rowe <readnews at sbcglobal.net>
  • Date: Wed, 30 Sep 2009 05:02:56 -0400 (EDT)

On 9/29/09 at 7:38 AM, jwmerrill at gmail.com wrote:

>Below is a definite integral that Mathematica does incorrectly.
>Thought someone might like to know:

>In[62]:= Integrate[Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}]

>Out[62]= \[Pi]/E

>What a pretty result--if it were true. The correct answer is
>\[Pi]*Cosh [1], which can be checked by adding a new parameter
>inside the argument of Cos and setting it to 1 at the end:

>In[61]:= Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity],
>\[Infinity]},
>Assumptions -> a \[Element] Reals]

>Out[61]= \[Pi] Cosh[a]

You didn't say what version of Mathematica you used to get the
above results. But given

Integrate[Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}]

Pi/E

E NIntegrate[Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}]

3.14159

Assuming[a \[Element] Reals,
  Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}]]

Pi E^-|a|

$Version

7.0 for Mac OS X x86 (64-bit) (February 19, 2009)

I am inclined to believe the result Pi/E is correct and the
result Pi Cosh[a] is in error.



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