Then as with and fixed
where denotes an arbitrary small positive constant.
This expansion is absolutely convergent for all finite , and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of as in .
If , with fixed, then as
and for ,
Sharp error bounds and an exponentially-improved extension for (8.11.7) can be found in Nemes (2016). This reference also contains explicit formulas for in terms of Stirling numbers and for the case an asymptotic expansion for as .
If and , then
For sharp error bounds and an exponentially-improved extension, see Nemes (2016). This reference also contains explicit formulas for the coefficients in terms of Stirling numbers.
For the function defined by (8.4.11),
With , an asymptotic expansion of follows from (8.11.14) and (8.11.16).
If is defined by
uniformly for , with
and as in §8.11(iii).