# §10.41 Asymptotic Expansions for Large Order

## §10.41(ii) Uniform Expansions for Real Variable

As through positive real values,

uniformly for . Here

10.41.7
10.41.8

where the branches assume their principal values. Also, and are polynomials in of degree , given by , and

10.41.9
.

For ,

10.41.10
10.41.11

For , , , see Bickley et al. (1952, p. xxxv).

For numerical tables of and the coefficients , , see Olver (1962, pp. 43–51).

## §10.41(iii) Uniform Expansions for Complex Variable

The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector , with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real -axis.

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).

 Figure 10.41.1: -plane. Symbols: : nonnegative integer, : complex variable, : points, : point, : point, : point, : points and : points Referenced by: §10.41(iii) Permalink: http://dlmf.nist.gov/10.41.F1 Encodings: pdf, png Figure 10.41.2: -plane. Symbols: Referenced by: §10.41(iii) Permalink: http://dlmf.nist.gov/10.41.F2 Encodings: pdf, png

For expansions in inverse factorial series see Dunster et al. (1993).

## §10.41(iv) Double Asymptotic Properties

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.

To establish (10.41.12) we substitute into (10.34.3), with and replaced by , by means of (10.41.13) observing that when is large the effect of replacing by is to replace , , and by , , and , respectively.

## §10.41(v) Double Asymptotic Properties (Continued)

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 .

It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when or equivalently . This is because and , do not form an asymptotic scale (§2.1(v)) as ; see Olver (1997b, pp. 422–425).