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Correspondence between direct and inverse Laplace transform?

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Hi,

Can anybody explain to me please why MATHEMATICA 10 returns the correct answer to the command:

LaplaceTransform[Exp[t]*Erfc[Sqrt[t]],t,s]

(the answer is 1/(Sqrt[s] +s) ) but it refuses to answer the command:

InverseLaplaceTransform[1/(Sqrt[s] +s),s,t]

I would appreciate any suggestions how to obtain the inverse transform in a possibly alternative way.

Leslaw

POSTED BY: Leslaw Bieniasz
Answer
7 months ago

Hi Lesław

I do not know what to write here, but from my experience it's normal behavior.Once LaplaceTransform it works and once InverseLaplaceTransform it does not work, or vice versa.Depending on the example.

Maybe someone from Wolfram development team answer your question and better explain you.

Try:

 InverseLaplaceTransform[1/(Sqrt[s] + s) // Apart, s, t]

Regards Mariusz.

POSTED BY: Mariusz Iwaniuk
Answer
7 months ago

Hi, If you use Simplify function, you can get original equation

InverseLaplaceTransform[1/(Sqrt[s] +s), s, t] // Simplify
POSTED BY: shahin eskandri
Answer
7 months ago

Dosen't work with :

InverseLaplaceTransform[BesselK[1, s], s, t] // Simplify
(* ? *)

but:

LaplaceTransform[(t HeavisideTheta[-1 + t])/Sqrt[-1 + t^2], t, s]
(* BesselK[1, s] *)
POSTED BY: Mariusz Iwaniuk
Answer
5 months ago

Here is an even simpler example:

LaplaceTransform[Exp[-t] Cos[2 t], t, s]

returns

(1 + s)/(4 + (1 + s)^2)

as expected. However,

InverseLaplaceTransform[(1 + s)/(4 + (1 + s)^2), s, t]

returns

1/2 E^((-1 - 2 I) t) (1 + E^(4 I t))
POSTED BY: Clark Wells
Answer
1 month ago
  InverseLaplaceTransform[(1 + s)/(4 + (1 + s)^2), s, t] // FullSimplify

returns:

 Exp[-t] Cos[2 t]

as expected.

POSTED BY: Mariusz Iwaniuk
Answer
1 month ago

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