       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]]
>
>

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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