I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$.
In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ F(\lambda)=\int_0^{\infty} e^{-\lambda t} f(t)dt. $$
By Paley-Wiener Theorem, we should show that
there exists $M$ s.t.
$$ \int_{-\infty}^{\infty} |F(x+iy)|^2 dy \leq M $$ for some $v>0$.
At this stage, i have a question : how to compute of $\int_{-\infty}^{\infty} |F(x+iy)|^2 dy \leq M $.
If $F(\lambda)=\frac{1}{\lambda}$, how can i compute that integration?