Riemann-Lebesgue Lemma

as if and

Proof

nahhhh

How quickly does this happen?

Integrating by parts:

Now assuming that or we find,
by the previous lemma on the integral term, that:

What if is 0 at both ends or the integration range is to

In this case, is small in all orders in as
In particular if

requiring as we have:

for all as long as is infinitely differentiable.