Regarding Laplace transformation and convolution integral

Question

I have a problem concerning Laplace transform and convolution integral as attached:

Problem statement Thanks.

Answer 1

If you define

$$ \mathcal{F}(s) = \int_{-\infty}^{+\infty}{\rm d}t~e^{-st}f(t) $$

then

$$ \int_{-\infty}^{+\infty}{\rm d}t~e^{-st/\tau}f(t) = \mathcal{F}(s/\tau) \not= \mathcal{F}(s) $$