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.