It turns out that for s in the interior of that interval (away from the endpoints), the function t → f(t)e-st will be integrable on the whole real line (you do not need to prove this). Therefore, the bilateral Laplace transform, (£f)(s) = f(t)e-* dt, is well defined for such s. Taking for granted that you can exchange derivatives in s and integrals in t (a version of Leibniz' rule for differentiation under the integral sign), give an expression for and then for, d" (Lf)(s) dsn 2 d" (Lf)(s) dsn mn := = (−1)n 1s=0 where n is an arbitrary natural number. Note that this question is about derivatives of the Laplace transform, not the Laplace transform of derivatives discussed in class in the unilateral case. "

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