Sunday, July 15, 2018
Inverse Laplace Transform of e^(2 s)/((s^2 + 1)
I recently needed to compute the inverse Laplace transform of:
$$\frac{e^{-2s}}{s^2 + 1}$$
For something like this we'd use the second time shifting theorem:
$$ \mathcal[L] f(t-a) u(t-a) = F(s) e^{-as} $$
where in this case $ F(s) = 1/(s^2 + 1) $
We just need to take the inverse transform of $ 1/(s^2+1) $ which is $ \sin (t) $ and then substitute in the time delay so that we arrive at:
$ \mathcal[L] f(t-a) u(t-a) = F(s) e^{-as} = u(t-2) \sin (t-2) $
Not too difficult. Out of curiously, I tried to use Mathematica to compute the inverse transform and most surprisingly it failed.
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