I'm looking at my old Differential Equations textbook and struggling with a sticky point regarding Laplace transforms.

He defines the Laplace transform in the standard way, saying that the Laplace transform is valid for all s such that the integral converges.

In looking at one of the exercises, he asks for the Laplace transform of the piecewise function

for

and 0 for

. I obtained the correct answer

, but the author says in his solutions manual that this is valid for s>0. Why not for s<0 in this case? Since the integral involved is not improper there should be no issue...am I missing something obvious?