Re: Incorrect symbolic improper integral
- To: mathgroup at smc.vnet.net
- Subject: [mg103630] Re: [mg103586] Incorrect symbolic improper integral
- From: "David Park" <djmpark at comcast.net>
- Date: Wed, 30 Sep 2009 05:03:55 -0400 (EDT)
- References: <30167826.1254225639354.JavaMail.root@n11>
Jason,
In Mathematica 7.0.1.0 I obtain for the second integral:
Integrate[Cos[a x]/(1 + x^2), {x, -\[Infinity], \[Infinity]},
Assumptions -> a \[Element] Reals]
% /. a -> 1
E^-Abs[a] \[Pi]
\[Pi]/E
Also, utilizing the symmetry about zero, this is the same answer given in
Gradshteyn & Ryzbik, 3.723, 2.
David Park
djmpark at comcast.net
http://home.comcast.net/~djmpark/
From: jwmerrill at gmail.com [mailto:jwmerrill at gmail.com]
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