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Tough Limit

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car...@colorado.edu

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Oct 9, 2008, 6:37:07 AM10/9/08
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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) ?

Jean-Marc Gulliet

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Oct 10, 2008, 4:35:29 AM10/10/08
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car...@colorado.edu wrote:

Perhaps by telling Mathematica that omega is real... since Mathematica
works over the field of complex numbers, by default.

In[9]:= Assuming[s > 0 && Im[\[Omega]] == 0,
Limit[Integrate[Sin[\[Omega]*t]*Exp[-s*t], {t, 0, x}],
x -> \[Infinity]]]

Out[9]= \[Omega]/(s^2 + \[Omega]^2)

Regards,
-- Jean-Marc

Peter Pein

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Oct 10, 2008, 4:36:33 AM10/10/08
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car...@colorado.edu schrieb:

Keeping in mind that
Integrate[f[x],{x,x0,Infinity}] is by def.
Limit[Integrate[f[x],{x,x0,x1},x1->Infinity],

Integrate[Sin[\[Omega]*t]*Exp[-s*t],{t,0,Infinity},Assumptions -> s>0]
or
LaplaceTransform[Sin[\[Omega]*t],t,s]

should (don't have Mathematica installed on this system) give the desired result.

Peter Pein

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Oct 10, 2008, 4:36:43 AM10/10/08
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Well, I guessed I missed sth.
Additionally, you have to assume a real omega:

Integrate[Sin[\[Omega]*t]*Exp[(-t)*s], {t, 0, Infinity},
Assumptions -> Inequality[s, Greater, 0, Equal, Im[\[Omega]]]]


or - because you'll get s^2+omega^2 in the denominator:

Integrate[Sin[\[Omega]*t]/E^(t*s), {t, 0, Infinity},
Assumptions -> s > Abs[Im[\[Omega]]]]

Peter

psycho_dad

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Oct 10, 2008, 4:38:40 AM10/10/08
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Just add some Assumptions for \[Omega] and s to the Limit function:

Limit[Integrate[Sin[\[Omega]*t]*Exp[-s*t], {t, 0, x}, Assumptions -> s

> 0], x -> \[Infinity], Assumptions -> \[Omega] > 0 && s > 0]

Cheers

Szabolcs Horvát

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Oct 10, 2008, 4:39:06 AM10/10/08
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You have to tell the assumptions to Limit too:

Limit[Integrate[Sin[\[Omega] t] E^(-s t), {t, 0, x},


Assumptions -> s > 0], x -> \[Infinity],

Assumptions -> s > 0 && \[Omega] \[Element] Reals]

To make things simpler and avoid duplication we could use Assuming:

Assuming[s > 0 && \[Omega] \[Element] Reals,
Limit[Integrate[Sin[\[Omega] t] E^(-s t), {t, 0, x}],
x -> \[Infinity]]]

David W.Cantrell

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Oct 10, 2008, 4:42:54 AM10/10/08
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In[2]:= Assuming[s > 0 && Element[\[Omega], Reals],
Limit[Integrate[Sin[\[Omega]*t]*Exp[(-s)*t], {t, 0, x}], x -> Infinity]]

Out[2]= \[Omega]/(s^2 + \[Omega]^2)

David

car...@colorado.edu

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Oct 11, 2008, 6:44:37 AM10/11/08
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> should (don't have Mathematica installed on this system) give the desired=
result.

Correct. The limit comes from a Laplace Transform homework exercise
for sophomores. Mathematica was allowed, along with three other CAS.
Students check their solution with tables.

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