ABSTRACT
In consequence of the Riemann-Lebesgue lemma (Chapter 3, $4.2) the last integral vanishes as n -, co. Therefore
9.3 This simple result is refinable in two important ways. In the first place it is unnecessary for f(t)-g(t) to be continuous on It I = r ; it is adequate that the integrals in (9.02) converge uniformly with respect to n. For example, Condition (ii) can be replaced by:
(iia) On the circle It 1 = r, the function f(t)-g(t) has afinite number of singularities and a t each singularity ti, say,
where uj is an assignable positive constant. The second refinement applies when f(t)-g(t) is m times continuously differ-
entiable on the circle It/ = r. In these circumstances the right-hand side of (9.02) may be integrated m times by parts to give the stronger result
also due to Darboux. And, again, this result is valid with the less restrictive condition
