# Inverse Laplace transform of a piecewise defined function

#### cg78ithaca

I understand the conditions for the existence of the inverse Laplace transforms are
$$\lim_{s\to\infty}F(s) = 0$$
and
$$\lim_{s\to\infty}(sF(s))<\infty$$
I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as
$$F(s) =\begin{cases} 1-s &\text{ if }0\le s\le1 \text{ and}\\ 0&\text{ if } s>1 \end{cases}$$
Clearly the limits above do satisfy the existence of the inverse condition, but I'm not sure how to determine the inverse.

I'm not sure whether the Bromwich integral method can be applied, since it would appear that if I choose gamma (the Browmich integral integration limits: gamma - i*Inf to gamma + i*Inf) between 0 and 1 the function to integrate is (1-s), whereas if I choose gamma > 1 then the Bromwich integral is obviously 0. I'm also not sure whether Post's inversion formula can be used since I'm not sure how to calculate high-order derivatives of a function which is piecewise-defined, in the limit when the order of differentiation --> Inf. Clearly for a finite k, the kth order derivative of F exists for all s except 1, but how about as k --> Inf?

Finally, just wondering if the two conditions I listed initially (the two limits) are sufficient for the inverse Laplace transform of F(s) to exist.

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