Re: Incorrect symbolic improper integral
- To: mathgroup at smc.vnet.net
- Subject: [mg103757] Re: Incorrect symbolic improper integral
- From: Mariano Suárez-Alvarez <mariano.suarezalvarez at gmail.com>
- Date: Mon, 5 Oct 2009 07:37:41 -0400 (EDT)
- References: <200909291138.HAA25632@smc.vnet.net> <h9vflr$e6v$1@smc.vnet.net>
On Oct 4, 6:36 am, d... at wolfram.com wrote:
> > On Sep 30, 8:33 am, Daniel Lichtblau <d... at wolfram.com> wrote:
> >> 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
>
> >> > [...]
>
> >> Pi/E is correct. For one thing, it agrees with NIntegrate. For another=
,
> >> you can find and verify correctness of an antiderivative, observe it
> >> crosses no branch cuts, and take limits at +-infinity to verify the
> >> definite integral.
>
> >> Moreover I do not replicate your parametrized result.
>
> >> In[20]:= Integrate[Cos[a*x]/(1+x^2), {x,-Infinity,Infinity},
> >> Assumptions -> Element[a,Reals]] // InputForm
> >> Out[20]//InputForm= Pi/E^Abs[a]
>
> >> I got that result, or something equivalent, in every Mathematica versi=
on
> >> I tried going back to 4. I may have missed some point releases. Also i=
t
> >> could be a timing-dependent problem, particularly if you are running
> >> version 6 (where it seems to be much slower than other versions).
>
> > What is a 'time-dependent problem' in this context?
>
> > -- m
>
> [...]
>
> As for what I mean by timing-dependent problems, there is brief mention i=
n
> "Symbolic definite integration: methods and open issues", which can be
> found here.
>
> http://library.wolfram.com/infocenter/Conferences/5832/
>
> I will quote from one of the notebooks:
>
> [...]
Ah. That makes a lot of sense. Thanks for the explanation!
-- m