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Homework Statement
Find the Laplace transform of
f(t) = t [itex]\forall[/itex] 0≤t≤T, 0 otherwise
Homework Equations
The Attempt at a Solution
I write the function as
[itex]tu(t)-t*u(t-T)[/itex]
That is turn on the function [itex]t[/itex] at [itex]t=0[/itex] and turn the function [itex]t[/itex] off at [itex]t=T[/itex]. It seems to be right to me.
But now I struggle with trying to take the Laplace transform of this. I know that [itex]L(u(t)) = \frac{1}{s}[/itex]. I know that [itex]L(f(t-T)) = e^{-sT}F(s)[/itex]. So I know that [itex]L(u(t-T)) = e^{-sT}\frac{1}{s}[/itex], but I'm not sure how to evaluate [itex]L(t*u(t-T))[/itex] because of the extra [itex]t[/itex] term, hence I'm stuck.
Thanks for any help.
I seem to be confused because [itex]F(s) = ∫_{0^{-}}^{∞}f(t)e^{-st}dt[/itex]. So I don't see how you can take the Laplace transform over a domain other than over [itex]0^{-}≤t≤∞[/itex], which this question seems to be asking me to do.
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