ABSTRACT
The last integral is absolutely and uniformly convergent, hence by Theorem 4.1 we have E,(x) = o(x-') as x -r OD.
Next, if all derivatives of q(t) are continuous in [a, b], then n integrations by parts yield
I(x) = 2 (kJ+' {eiaxq(;4(a) -e ~ ~ ~ ~ ( ~ ) ( b ) } + E,, (x), 3-0
where
Again, e,,(x) = o(x-") by the Riemann-Lebesgue lemma. Hence (5.04) furnishes an asymptotic expansion of I(x) for large x.' 5.2 These results extend easily to an infinite range of integration. Suppose that all derivatives of q(t) are continuous in [a, a) and each of the integrals
J a converges uniformly for all sufficiently large x. Letting b -r og in (5.02) and (5.03) we see that eibXq(b) must tend to a constant limiting value, and since x can take more than one value it follows that q(b) -, 0 as b -, OD. Application of Theorem 4.1 then shows that
This argument may be repeated successively for n = 1,2, ... in (5.04) and (5.05). In this way we establish .
