# §10.19 Asymptotic Expansions for Large Order

## §10.19(ii) Debye’s Expansions

For error bounds for the first of (10.19.3) see Olver (1997b, p. 382).

## §10.19(iii) Transition Region

As , with fixed,

with sectors of validity . Here and are the Airy functions (§9.2), and

10.19.10
10.19.11

with sectors of validity and , respectively. Here

10.19.14
10.19.15

For proofs and also for the corresponding expansions for second derivatives see Olver (1952).

For higher coefficients in (10.19.8) in the case (that is, in the expansions of and ), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). The last reference also includes the corresponding expansions for and .