[Math] How does one find the Laplace transform for the product of the Dirac delta function and a continuous function

Question

As an example, what is the Laplace transform for the following:
$$g(t)=\delta(t-2\pi) cos t$$

I've worked through a few examples that required finding $\mathcal{L}\{\delta(t-t_0)\}=e^{-st_0}$, but I'm completely stumped when it comes to finding $\mathcal{L}\{\delta(t-t_0)f({t})\}$. Is there a general formula for such a case?

Answer 1

Here is how (check Laplace transform of $\delta(t-a)$)

$$ \int_{0}^{\infty} g(t)e^{-st} dt = \int_{0}^{\infty}\delta(t-2\pi) \cos t\, e^{-st}\,dt = \cos(2\pi)e^{-2\pi s }= e^{-2\pi s }. $$