If through positive real values with fixed, then
As through positive real values,
uniformly for . Here
where the branches assume their principal values. Also, and are polynomials in of degree , given by , and
For , , , see Bickley et al. (1952, p. xxxv).
For numerical tables of and the coefficients , , see Olver (1962, pp. 43–51).
Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the -plane and the -plane. The curve in the -plane is the upper boundary of the domain depicted in Figure 10.20.3 and rotated through an angle . Thus is the point , where is given by (10.20.18).
For derivations of the results in this subsection, and also error bounds, see Olver (1997b, pp. 374–378). For extensions of the regions of validity in the -plane and extensions to complex values of see Olver (1997b, pp. 378–382).
For expansions in inverse factorial series see Dunster et al. (1993).
In the case of (10.41.13) with positive real values of the result is a consequence of the error bounds given in Olver (1997b, pp. 377–378). Then by expanding the quantities , , and , , and rearranging, we arrive at an expansion of the right-hand side of (10.41.13) in powers of . Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with replaced by , up to and including the term in . It also enjoys the same sector of validity.
Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. We first prove that for the expansions (10.20.6) for the Hankel functions and the -asymptotic property applies when , respectively. This is a consequence of the error bounds associated with these expansions. We then extend the validity of this property from to in the sector in the case of , and to in the sector in the case of . This is done by re-expansion with the aid of (10.20.10), (10.20.11), and §10.41(ii), followed by comparison with (10.17.5) and (10.17.6), with replaced by . Lastly, we substitute into (10.4.4), again with replaced by . The final results are:
as in , or equivalently as in , for fixed and fixed .