I wonder why Fourier and Laplace transforms were dismissed very early on? It is very tempting to generalize the the differentiation property to non-integer n an

I wonder why Fourier and Laplace transforms were dismissed very early on? It is very tempting to generalize the the differentiation property to non-integer n and do a inverse transform to get any non-integer derivative. For instance, I would imagine that f^(1/2) = L^{-1}[ sqrt(s) L[f] ]. Is there a particular difficulty in doing the integral?