Re: Incorrect symbolic improper integral
- To: mathgroup at smc.vnet.net
- Subject: [mg103604] Re: [mg103586] Incorrect symbolic improper integral
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Wed, 30 Sep 2009 04:58:59 -0400 (EDT)
- References: <200909291138.HAA25632@smc.vnet.net>
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
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 version
I tried going back to 4. I may have missed some point releases. Also it
could be a timing-dependent problem, particularly if you are running
version 6 (where it seems to be much slower than other versions).
Daniel Lichtblau
Wolfram Research
- References:
- Incorrect symbolic improper integral
- From: "jwmerrill@gmail.com" <jwmerrill@gmail.com>
- Incorrect symbolic improper integral