Re: Incorrect symbolic improper integral
- To: mathgroup at smc.vnet.net
- Subject: [mg103608] Re: Incorrect symbolic improper integral
- From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
- Date: Wed, 30 Sep 2009 04:59:47 -0400 (EDT)
- References: <h9srop$p7s$1@smc.vnet.net>
"jwmerrill at gmail.com" <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.
It is, actually. Instead, it is your result with the parameter, below,
which is incorrect.
> 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]
Here's a correct result from version 5.1:
In[7]:= Integrate[Cos[a x]/(1 + x^2), {x, -Infinity, Infinity},
Assumptions -> Element[a,Reals]]
Out[7]= Pi/E^Abs[a]
David