Correspondence between direct and inverse Laplace transform?

Questions
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1

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

2

Hi Les?aw

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.

3

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

InverseLaplaceTransform[1/(Sqrt[s] +s), s, t] // Simplify
4

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] *)
5

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))
6

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

 Exp[-t] Cos[2 t]

as expected.