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## Homework Statement

Using the t-shifting theorem, find the laplace transform of

[itex] f(x) = tu(t-\pi)[/itex]

## Homework Equations

[itex] L[f(t-a)u(t-a)] = F(s)e^{-as}[/itex]

## The Attempt at a Solution

Now firstly I should state I already know the answer to the problem, the issue is getting to said answer.

I think if I understood what [itex] f(x) = tu(t-\pi)[/itex] actually looked like it might help, but here's my process.

[itex] a = \pi[/itex] so using the relationship between [itex]u(t-a) = e^{-as}[/itex] I get [itex]e^{-as} = e^{-\pi s}[/itex].

f(t) = t, the laplace transformation of t is [itex]\frac{n!}{s^{n+1}} = \frac{1}{s^2}[/itex]

Now multiplying [itex]e^{-\pi s}[/itex] by F(s) gives:

[itex]\frac{e^{-\pi s}}{s^2}[/itex]

So that's as far as I get, the problem is, the answer includes an extra term [itex]\frac{\pi e^{-\pi s}}{s}[/itex] and I have no idea how the get it. So the complete answer is [itex]\frac{e^{-\pi s}}{s^2} + \frac{\pi e^{-\pi s}}{s}[/itex]

I feel like I've missed something extremely basic, but no matter how many YouTube videos I watch or different text books I read, I can't make heads or tails of it.. thanks!