ca.analysis and odes - inverse Laplace transform of $delta_1(cdot)$

Let's try to find a function $psi(x)$ such that for Laplace transform $tilde{f}(p)=int_0^{infty} f(y) e^{-py} dy$ one has $f(x)=int_0^{infty} tilde{f}(p)psi(px)dp$ (here we do not specify classes of functions, for which this should hold).

In other words, $tilde{psi}(p)=delta_1(p)$ in the sense of distributions.

Or, Stiltjes transform $int_0^{infty} frac{psi(t)}{t+y} dt=e^{-y}, y>0$.

(just represent $frac1{t+y}=int_0^{infty} e^{-q(t+y)}dq$ and change order of integration).

So the question(s) is(are):

Does such function exist, if it exists, what are its properties, where is it written about all this stuff and so on.