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Re: laplace transform
*To*: mathgroup at smc.vnet.net
*Subject*: [mg65144] Re: laplace transform
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Wed, 15 Mar 2006 06:30:36 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <dv68hj$nvj$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <dv68hj$nvj$1 at smc.vnet.net>,
Marlies.Goorden at physics.unige.ch wrote:
> I have a problem with the Laplace transform of mathematica. I
> want to know the laplace transform of sin(a*t).
> When I type
> LaplaceTransform[sin(a*t),t,s]
> mathematica gives me
> \sqrt(a^2) sign(a)/(a^2+s^2)
>
> On the other hand my mathematics books gives the answer
> a/(s^2+a^2)
>
> For complex a the answer is not the same. If I choose for
> example
> a=0.3+0.5i and
> s=1
> the two formulas give me
> -0.096+0.65i and 0.51+0.41i respectively.
>
> A numerical integration,
> i.e. NIntegrate[Sin((0.3+0.5i)*t)*Exp[-t],{t,0,Infinity}]
> gives me the same numerical value as the mathematics book
> formula.
> Is the mathematica formula wrong?
Yes and no. LaplaceTransform is making assumptions that are incompatible
with a being complex. If you enter
LaplaceTransform[Sin[a t], t, s,
GenerateConditions -> True, Assumptions -> NotElement[a,Reals]]
you get a/(s^2+a^2). Alternatively, if you put a -> x + I y then you
will get the answer that you desire:
LaplaceTransform[Sin[(x + I y) t], t, s]
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
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