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Does the InverseLaplaceTransform work correctly?

Leslaw Bieniasz
Posted 5 years ago

Hi, I notice a strange thing: 

InverseLaplaceTransform[(g+s)/Sqrt[s],s,t]

yields a result proportional to (2gt-1)/t^(3/2), with a coefficient involving HeavisideTheta function. Is this a correct result? In principle, the term 1/t^(3/2) does not have a Laplace transform, as it is not integrable at t->0. I do not see how the HeavisideTheta function might change this fact. In addition, if we take g=0, then (g+s)/Sqrt[s] reduces to Sqrt[s], for which MATHEMATICA currently says there is no inverse transform, although older versions returned some expressions proportional to 1/t^(3/2). So, does the InverseLaplaceTransform operate correctly? It would be nice to trust MATHEMATICA, but after such experiences I am no longer sure if I can. Leslaw
POSTED BY: Leslaw Bieniasz
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Contact support here. they are very responsive about suspected bugs.

Regards

POSTED BY: Neil Singer

Is there any way to inform MATHEMATICA developers about this problem? Otherwise the error will persist in future versions. Leslaw
POSTED BY: Leslaw Bieniasz

Yes, I do agree (you got a 'like' from me!). The Laplace transform of $\frac{1}{t^{3/2}}$ does not exist. And it should not make any difference whether a function f[t] is transformed or f[t] HeavisideTheta[t] and the like. But in fact we find:

form = Inactivate[LaplaceTransform[1/t^(3/2) #, t, s], 
     LaplaceTransform] & /@ {1, HeavisideTheta[t], Sign[t]};
Grid[{#, Activate[#]} & /@ form, Frame -> All]

enter image description here

Strange indeed!

POSTED BY: Henrik Schachner

Well, for me this only proves that both LaplaceTransform[] and InverseLaplaceTransform[] make the same mistake. If we consider the expression (g+s)/Sqrt[s] as a sum of g/Sqrt[s] and Sqrt[s] than the inverse of Sqrt[s] obviously does not exist. Also, if we consider (gt-1)/t^(3/2) as a sum of g/Sqrt[t] and -1/t^(3/2), then the Laplace transform of 1/t^(3/2) does not exist, because Exp[-st]/t^(3/2) is not integrable close to t=0. I am trying transforming and retransforming various other expressions and I am beginning to suspect that both LaplaceTransform[] and InverseLaplaceTransform[] produce nonsense without any warning.

Les?aw
POSTED BY: Leslaw Bieniasz

I personally find it convincing that the Laplace transform of your inverse transform results in your original function. We have:

enter image description here

Now the term

(-1 + 2 HeavisideTheta[-t] - HeavisideTheta[-t, t])

is identical to (-Sign[t]), and then

Simplify[LaplaceTransform[-(((-1 + 2 g t) (-Sign[t]))/(
   2 Sqrt[\[Pi]] t^(3/2))), t, s], Assumptions -> s \[Element] Reals]

gives the function you started with.
POSTED BY: Henrik Schachner
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