Suppose that f: (0, ) R is continuous and bounded and that £{f} exists at some a (0, ). Let a) Prove that for all

Suppose that f: (0, ˆž) †’ R is continuous and bounded and that £{f} exists at some a ˆˆ (0, ˆž). Let

a) Prove that

for all N ˆˆ N.
b) Prove that the integral ˆ«0ˆž e-(s-a)t(ɸ)(t)dt converges uniformly on [b, ˆž) for any b > a and

c) Prove that £{f} exists, is continuous on (a, ˆž), and satisfies

d) Let g(t) = tf(t) for t ˆˆ (0, ˆž). Prove that £{f} is differentiate on (a, ˆž) and

for all s ˆˆ (a, ˆž).
e) If, in addition, fʹ is continuous and bounded on (0, ˆž), prove that

limo dif}(s) =0. L(f')(s) = sL(f)(s) _ f (0)