MATH319 Slides

77 Differentiating Laplace transforms

(iii) Let s>β+δ for some δ>0 and consider -δ<h<δ. Note that eδxe-sxf(x) is integrable, and xeδx/δ, so xf(x) also satisfies (E). Also

e-(s+h)x-e-sxh=e-sx(e-hx-1h)-xe-sx

as h0. Hence

(f)(s+h)-(f)(s)h=0e-(s+h)x-e-sxhf(x)𝑑x
-0e-sxxf(x)𝑑x.

To make this precise, we consider