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