In general is there a way to know is some function is a Laplace transform or not?
First I tried to find some known laplace transforms which had just a $s$ in the numerator and then tried to manipulate the inverse function. For eg. $\cos{t}$, $t\sin{t}$, but couldnt do it.
Next I tried to do ab initio using the definite integral definition of laplace from $0$ to $\infty$. So I assumed the integral answer as as $F(s,t)$ and then I took $F(s,\infty)-F(s, 0)=s$. But no such F comes to my mind… any help is appreciated.
$s$ is the Laplace transform of the derivative of the $\delta$-function, see for example this question, which is a distribution and not a function. I am not aware of the simple characterisation of the range of Laplace transform.