again denoting an arbitrary small positive constant. Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii).
For an exponentially-improved asymptotic expansion of see §2.11(iii).
For and let and define ,
so that is a polynomial in of degree when . In particular,
uniformly for .