Re: Incorrect symbolic improper integral
- To: mathgroup at smc.vnet.net
- Subject: [mg103624] Re: [mg103586] Incorrect symbolic improper integral
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Wed, 30 Sep 2009 05:02:45 -0400 (EDT)
- References: <200909291138.HAA25632@smc.vnet.net>
The answer returned by Integrate agrees with the one given by
NIntegrate, which uses very different methods:
Integrate[
Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}] // N
1.15573
NIntegrate[
Cos[x]/(1 + x^2), {x, -\[Infinity], \[Infinity]}] // N
1.15573
Simple numerical checks show that your proposed answer is far too
large and can't be right. And what is even more curious is that my
Mathemaica 7.01 returns:
Integrate[Cos[a*x]/(1 + x^2), {x, -Infinity, Infinity},
Assumptions -> Element[a, Reals]]
Pi/E^Abs[a]
Exactly the same answer is returned by all versions of Mathematica
from 5.2. and 6.03 (the only ones I have tested). So which version
gave your answer?
Andrzej Kozlowski
On 29 Sep 2009, at 20:38, 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]
>
> Regards,
>
> Jason Merrill
>
- References:
- Incorrect symbolic improper integral
- From: "jwmerrill@gmail.com" <jwmerrill@gmail.com>
- Incorrect symbolic improper integral