Laplace transform

[Kroldar]

Let
$x(t) = {([t], t>0),(0, t<0):}$

Calculate Laplace transform of $x(t)$.

[Kroldar]

Of course, this exercise was not so difficult and elgiovo's solution is correct.

I posted it to introduce a nice theorem:
Let $x_0(t) in L_(loc)^1(RR)$ and $x_0(t) = 0 AA t < 0$. In these hypothesis the series
$sum_(k=0)^(+oo) x_0(t-ktau)$
converges a.e. in $RR$. If we call $x(t)$ the sum of the series, it results
$ccL_u[x(t)] = (ccL[x_0(t)])/(1-e^(-stau))$

So, we can calculate unilateral Laplace-transform of shifted sums.

[elgiovo]

"Camillo":
I suppose you have used $[t] = sum_(k=1)^(+oo) H(t-k )$ and then trasformed each addendum etc ....

Yes, right assumption. But maybe Mr. Kroldar knows a better solution...

[Camillo]

"elgiovo":What about $X(s)=(e^(-s))/s sum_(k>=0) e^(-ks)$? It is a simple geometric series, whose sum is $X(s)=1/(s(e^s-1))=e^(-s)/(2s)(1+"coth"s/2)$.

I suppose you have used $[t] = sum_(k=1)^(+oo) H(t-k )$ and then trasformed each addendum etc ....

[Camillo]

"Kroldar":[quote="Camillo"]It is $X(s) = 1/s^2 $ with abscissa of absolute convergence $= 0 $.

$[t]$ is not $t$... $[t]$ is the function that in Italian we call "parte intera di $t$".

Does anyone know a nice and very simple way to calculate this transform?[/quote]

I had a doubt .....that $ [t] $ is not $t $ .

[elgiovo]

What about $X(s)=(e^(-s))/s sum_(k>=0) e^(-ks)$? It is a simple geometric series, whose sum is $X(s)=1/(s(e^s-1))=e^(-s)/(2s)(1+"coth"s/2)$.

[Kroldar]

"Camillo":It is $X(s) = 1/s^2 $ with abscissa of absolute convergence $= 0 $.

$[t]$ is not $t$... $[t]$ is the function that in Italian we call "parte intera di $t$".

Does anyone know a nice and very simple way to calculate this transform?

[Camillo]

It is $X(s) = 1/s^2 $ with abscissa of absolute convergence $= 0 $.