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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




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