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Re: laplace transform


Marlies.Goorden at physics.unige.ch wrote:

>Hi,
>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?
>Thank you for your help,
>Marlies Goorden
>
>  
>
Possibly the reason for the Sign[a] might be that the Laplace transform 
has a switch at the origin depending on the sign of a
In[1]:=
Plot[With[{a=1},LaplaceTransform[Sin[a*t],t,s]],{s,-10,10}]
Plot[With[{a=-1},LaplaceTransform[Sin[a*t],t,s]],{s,-10,10}]
Assuming[a>0,LaplaceTransform[Sin[a*t],t,s]]//InputForm
Assuming[a<0,LaplaceTransform[Sin[a*t],t,s]]//InputForm


Out[1]=
â??Graphicsâ??

Out[2]=
â??Graphicsâ??

Out[4]//InputForm=
a/(a^2 + s^2)

As to the complex vals, I don't see a problem

In[5]:=
With[{a=0.3+0.5*I,s=1},LaplaceTransform[Sin[a*t],t,s]]//N
Integrate[Sin[(0.3+0.5*I)*t]*Exp[-t],{t,0,â??}]

Out[5]=
0.505279\[InvisibleSpace]+0.414781 \[ImaginaryI]

Out[6]=
0.505279\[InvisibleSpace]+0.414781 \[ImaginaryI]

Hope this helps

Pratik


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