Laplace Transform of (t - 120) u(t - 120)

Question

I am puzzled by what should be a simple problem. I am deriving the Laplace transform for

$$ (t-120) u(t-120) $$

where $u(t)$ is the unit step signal. Using the second shift theorem, ie

$$ \mathcal{L} (f(t-a) u(t-a) ) = e^{-as} F(s) $$

I would have thought that:

$$ \mathcal{L} ((t-120) u(t-120) = e^{-120 s} \left( \frac{1}{s^2} - \frac{120}{s} \right) $$

To check I ran the same problem through Mathematica 10 and it gave:

$$ \mathcal{L} ((t-120) u(t-120) = \frac{e^{-120 s}}{s^2} $$

which is different. Where have I gone wrong?

Answer 1

The thing is that you have to identify $f(t)=t$. Then you have $F(s)=1/s^2$. If you consider $f(t)=t-120$, where is your $f(t-a)$? It would be $t-a-120$, and you can't apply the theorem. However if $f(t)=t$ then your problem ($(t-120)\,\mathcal U(t-120)$) can be expressed as $f(t-a)\,\mathcal U(t-a)$, where $a=120$, and now you can apply the theorem.

So what you are doing is to take the laplace transform of $120$; don't, because you aren't using the theorem anymore.