Another class of transform is that of the Laplace transform
and it appears in methods of solution of linear differential equations and in
the relationship between the partition function Q(T) and the corresponding
density of states
Finding the inverse of the Laplace transform is not simple in general and tables of transform pairs are available. Of course one could just consult Mathematica
Needs["Calculus`LaplaceTransform`"] les = LaplaceTransform[Exp[-5 x], x, s] 1 ----- 5 + s InverseLaplaceTransform[les,s,x] -5 x E
Some of the results below have application to the damped oscillator problem which is governed by a differential equation
<<Calculus`LaplaceTransform`
LaplaceTransform[Cosh[t], t, s]
s
-------
2
-1 + s
LaplaceTransform[Sinh[t], t, s]
1
-------
2
-1 + s
InverseLaplaceTransform[1/(k+s), s, t]
-(k t)
E
InverseLaplaceTransform[1/(s), s, t]
1
InverseLaplaceTransform[1/(s(s+k)), s, t]
1 1
- - ------
k k t
E k
InverseLaplaceTransform[1/(s^2 + eom*s + kom), s, t]
2
-(eom t)/2 - (Sqrt[eom - 4 kom] t)/2
E
-(--------------------------------------) +
2
Sqrt[eom - 4 kom]
2
-(eom t)/2 + (Sqrt[eom - 4 kom] t)/2
E
--------------------------------------
2
Sqrt[eom - 4 kom]
InverseLaplaceTransform[s/(s^2 + eom*s + kom), s, t]
2
((-eom + Sqrt[eom - 4 kom]) t)/2 2
-(E (eom - Sqrt[eom - 4 kom]))
---------------------------------------------------------------- +
2
2 Sqrt[eom - 4 kom]
2
eom + Sqrt[eom - 4 kom]
------------------------------------------------------
2
((eom + Sqrt[eom - 4 kom]) t)/2 2
2 E Sqrt[eom - 4 kom]
InverseLaplaceTransform[1/(s+1)(s-1), s, t]
-2
-- + DiracDelta[t]
t
E