Lets suppose to 'discharge' the fact that the function is 'truncated' and consider only the function . In that case is...
The result (1) is valid only if the integral converges and that is true for ...
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?