Re: Incorrect symbolic improper integral
- To: mathgroup at smc.vnet.net
- Subject: [mg103621] Re: Incorrect symbolic improper integral
- From: pfalloon <pfalloon at gmail.com>
- Date: Wed, 30 Sep 2009 05:02:12 -0400 (EDT)
- References: <h9srop$p7s$1@smc.vnet.net>
On Sep 29, 9:41 pm, "jwmerr... at gmail.com" <jwmerr... 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
I think the first result is correct. For the second integral, (your Out
[61]), I get a different result to you -- one that is consistent with
the original integral:
In[119]:= Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \
[Infinity]}, Assumptions -> a \[Element] Reals]
Out[119]= E^-Abs[a] \[Pi]
In[120]:= % /. a->1
Out[120]= \[Pi]/E
In[121]:= $Version
Out[121]= 7.0 for Microsoft Windows (32-bit) (February 18, 2009)
Maybe your second result is a bug from an older version of Mathematica
that has now been fixed?
Also, doing the indefinite integral then taking limits gives the same
result, and differentiating the indefinite integral recovers the
integrand, all of which support the original integral being correct:
In[122]:= f[x_] = Integrate[Cos[x]/(1 + x^2), x];
In[123]:= D[f[x], x] == Cos[x]/(1+x^2) // FullSimplify
Out[123]= True
In[126]:= Limit[f[x] - f[-x], x->Infinity]
Out[126]= \[Pi]/E
Cheers,
Peter.