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