Re: Tough Limit
- To: mathgroup at smc.vnet.net
- Subject: [mg92661] Re: [mg92644] Tough Limit
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Fri, 10 Oct 2008 04:31:56 -0400 (EDT)
- References: <200810091036.GAA24087@smc.vnet.net>
On 9 Oct 2008, at 19:36, carlos at colorado.edu wrote:
> How can I get
>
>
> Limit[Integrate[Sin[\[Omega]*t]*Exp[-s*t],{t,0,x},
> Assumptions->s>0],x->\[Infinity]]
>
> to answer \[Omega]/(\[Omega]^2+s^2) ?
>
First, the answer you give is not correct without the assumption that
Omega is real. For example, take Omega = 2Pi I and take s= 2Pi and you
will easily see that the integral does not converge. So assuming that
Omega is real you get:
FullSimplify[Limit[
Integrate[Sin[\[Omega]*t]/
E^(s*t), {t, 0, x}],
x -> Infinity,
Assumptions -> s > 0 &&
Im[\[Omega]] == 0],
Assumptions -> s > 0 &&
Element[\[Omega], Reals]]
\[Omega]/(s^2 + \[Omega]^2)
or, more simply:
Integrate[Sin[\[Omega]*t]/E^(s*t),
{t, 0, Infinity},
Assumptions -> s > 0 &&
Im[\[Omega]] == 0]
\[Omega]/(s^2 + \[Omega]^2)
Andrzej Kozlowski
- References:
- Tough Limit
- From: carlos@colorado.edu
- Tough Limit