(a) Let f : R -+ R and g : R - R be functions whose Laplace transforms LIf (x)] = F(p) and Lig(r)] = G(p) exist for p > 0. Moreover, assume these functions satisfy the conditions . f'(x) = g(x) for all r E R, and . lime-too e pi f(x) = 0 for all p > 0. Show that F(p) = =(G(p) + f(0)) for all p > 0. (* ) P (b) Verify that formula (*) holds in the case where f(r) = cos(ar) and g(I) = f'(x)