# §10.20 Uniform Asymptotic Expansions for Large Order

## §10.20(i) Real Variables

Define to be the solution of the differential equation

that is infinitely differentiable on the interval , including . Then

10.20.3,

all functions taking their principal values, with , corresponding to , respectively.

uniformly for in all cases, where and are the Airy functions (§9.2).

In the following formulas for the coefficients , , , and , , are the constants defined in §9.7(i), and , are the polynomials in of degree defined in §10.41(ii).

10.20.10
10.20.11
10.20.12
10.20.13

### ¶ Interval

In formulas (10.20.10)–(10.20.13) replace , , , and by , , , and , respectively.

Note: Another way of arranging the above formulas for the coefficients , and would be by analogy with (12.10.42) and (12.10.46). In this way there is less usage of many-valued functions.

### ¶ Values at

10.20.14

Each of the coefficients , , , and , , is real and infinitely differentiable on the interval . For (10.20.14) and further information on the coefficients see Temme (1997).

For numerical tables of , and , , , and see Olver (1962, pp. 28–42).

## §10.20(ii) Complex Variables

The function given by (10.20.2) and (10.20.3) can be continued analytically to the -plane cut along the negative real axis. Corresponding points of the mapping are shown in Figures 10.20.1 and 10.20.2.

The equations of the curved boundaries and in the -plane are given parametrically by

10.20.15,

respectively.

The curves and in the -plane are the inverse maps of the line segments

10.20.16,

respectively. They are given parametrically by

where is the positive root of the equation . The points where these curves intersect the imaginary axis are , where

10.20.18

The eye-shaped closed domain in the uncut -plane that is bounded by and is denoted by ; see Figure 10.20.3.

 Figure 10.20.1: -plane. and are the points . . Symbols: : nonnegative integer, : complex variable, : point, : point, : points, : points, : points, : points, : coefficients, and : point Referenced by: §10.20(ii) Permalink: http://dlmf.nist.gov/10.20.F1 Encodings: pdf, png Figure 10.20.2: -plane. and are the points Symbols: : base of exponential function, : solution and : points Referenced by: §10.20(ii) Permalink: http://dlmf.nist.gov/10.20.F2 Encodings: pdf, png
Figure 10.20.3: -plane. Domain (unshaded). .

As through positive real values the expansions (10.20.4)–(10.20.9) apply uniformly for , the coefficients , , , and , being the analytic continuations of the functions defined in §10.20(i) when is real.

For proofs of the above results and for error bounds and extensions of the regions of validity see Olver (1997b, pp. 419–425). For extensions to complex see Olver (1954). For resurgence properties of the coefficients (§2.7(ii)) see Howls and Olde Daalhuis (1999). For further results see Dunster (2001a), Wang and Wong (2002), and Paris (2004).

## §10.20(iii) Double Asymptotic Properties

For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of see §10.41(v).