Does Fourier imply Laplace? Can we find a function $f(t)$ for which $$\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt,$$ converges but $$\int_{-\infty}^{+\infty}f(t)e^{-st}dt,$$ does not ?
Here, $j^2=-1$, $\omega$ is a real number and $s$ is a complex number.
I am thinking that we can find such an $f(t)$, for example, when $Re(s)>0$.

$f(t)=\frac 1 {1+t^{2}}$ is such a function.
