Re: Tough Limit

[Bob Hanlon]

Also

Assuming[{s > 0, Element[w, Reals]},
Limit[Integrate[Sin[w*t]*Exp[-s*t], {t, 0, x}],
x -> Infinity]]

w/(s^2 + w^2)

Bob Hanlon

---- Bob Hanlon <han...@cox.net> wrote:

=============
Integrate[Sin[w*t]*Exp[-s*t], {t, 0, Infinity},
Assumptions -> {s > 0, Element[w, Reals]}]

w/(s^2 + w^2)

Bob Hanlon

---- car...@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) ?

--

Bob Hanlon

--

Bob Hanlon

[Andrzej Kozlowski]

On 9 Oct 2008, at 19:36, car...@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

[Bob Hanlon]

Integrate[Sin[w*t]*Exp[-s*t], {t, 0, Infinity},
Assumptions -> {s > 0, Element[w, Reals]}]

w/(s^2 + w^2)

Bob Hanlon

---- car...@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) ?

--

Bob Hanlon