Originally Posted by

**jenjen** Problem #4: find the inverse Laplace Transform of the function $\displaystyle

e^{-10s}\frac{2}{s^{2}+4}

$

First thing we need is the convolution theorem for Laplace transforms:

$\displaystyle

\mathcal{L}^{-1} f(s)g(s) = \int_0^t F(u)G(t-u)\ du

$

Then we need the following:

$\displaystyle

\mathcal{L}^{-1} e^{-as}=\delta (t-a)

$,

and:

$\displaystyle

\mathcal{L}^{-1} \frac{1}{s^2+a^2}=\frac{\sin(at)}{a}

$.

So putting this all together we have:

$\displaystyle

\mathcal{L}^{-1} e^{-10s}\frac{2}{s^2+4}=\int_0^t \delta (u-10) \sin(2(t-u)) \ du

$,

which from the definition of the $\displaystyle \delta$ functional gives:

$\displaystyle

\mathcal{L}^{-1} e^{-10s}\frac{2}{s^2+4}=\sin(2(t-10))\ \ t>10

$

$\displaystyle

=0\ \ t<10

$

This also follows directly form the translation theorem.

RonL