If through positive real values, with fixed, then
If through positive real values with fixed, then
If through positive real values with fixed, and
In these expansions and are the polynomials in of degree defined in §10.41(ii).
As , with fixed,
with sectors of validity . Here and are the Airy functions (§9.2), and
with sectors of validity and , respectively. Here
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 .
See also §10.20(i).