It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace Transform of a positive function of form $s^{-\alpha}$?


The answer is trivial when $\alpha>0$, because $\mathcal{L}\{t^{\alpha-1}\}=\Gamma(\alpha) s^{-\alpha}$ when $\alpha>0$.


But and if $\alpha<0$? The exact function is not known by any people, but could we say something about positiveness?


Thank you so much.



Is inverse Laplace Transform of a power of $s$ a positive function?
The inverse Laplace transform of $s$ is the derivative $\delta'(x)$ of the Dirac delta function (a distribution). This is not a positive "function", as you can check by evaluating the integral with a test function $f(x)$,
$$\int\_{-\infty}^\infty f(x)\delta'(x)\,dx=-\int\_{-\infty}^\infty f'(x)\delta(x)\,dx=-\lim\_{x\rightarrow 0}f'(x).$$
So even if $f(x)\geq 0$ for all $x$, this integral will be negative if $f'(0)>0$.


The same applies to higher integer powers of $s$, the inverse Laplace transform of $s^p$ is the $p$-th derivative of a delta function, which is not a positive function, in the sense that the integral with a positive test function can be negative,
$$\int\_{-\infty}^\infty f(x)\delta^{(p)}(x)\,dx=(-1)^p\int\_{-\infty}^\infty f^{(p)}(x)\delta(x)\,dx=-\lim\_{x\rightarrow 0}f^{(p)}(x).$$


