Laplace of two functions multiplied together?

Question

I'm trying to figure out how to do the Laplace transform of $\delta(t-\pi) \sin{t}$.

I know the Laplace transform of each of these. But how would I find it when they are multiplied together?

Answer 1

The problem here is that

$$\int_0^{\infty} dt \: \delta(t-\pi) \, \sin{t}\, e^{-s t} = \sin{\pi} \, e^{-s \pi} = 0$$

Answer 2

In general, $f(t)\delta(t -a ) = f(a)\delta(t -a )$ if $f(t)$ is continuous at $t = a$. So in the case at hand $\sin(t)\delta(t - \pi) = \sin(\pi)\delta(t - a) = 0$. So you're just taking the Laplace transform of the zero function, which is again zero.