How do you compute the Laplace Transform of [[x]] - I think that it has something to do with step functions

How do you compute the Laplace Transform of [[x]] - I think that it has something to do with the greatest integer function.

L { [[x]] } = ???


*****Thanks for all of your help!!!! I really appreciate it!!!*****

Just use the definitions and write down the first few integrals:

  1
 ∫ 0 e^(-st) dt +
0

  2
 ∫ 1 e^(-st) dt +
1

  3
 ∫ 2 e^(-st) dt +
2

  4
 ∫ 3 e^(-st) dt + ...
3

You should get

0 + (1/s)[e^(-s) - e^(-2s) + 2e^(-2s) - 2e^(-3s) + 3e^(-3s) - 3e^(-4s) +- ...]

which simplifies to

(1/s)[e^(-s) + e^(-2s) + e^(-3s) + e^(-4s) + ,,,]

The series in the square brackets is an infinite geometric sum with a = r = e^(-s) , which converges to e^(-s)/(1 - e^(-s)) for e^(-s) < 1; that is, for s > 0. Multiply the numerator and denominator by e^s to get the simpler form 1/(e^s - 1). Thus,

L { [[x]] } = 1/(s(e^s - 1))

.

Ans

1/s^2