§10.19 Asymptotic Expansions for Large Order
Contents
§10.19(i) Asymptotic Forms
If
through positive real values, with
fixed, then
10.19.1
10.19.2
§10.19(ii) Debye’s Expansions
If
through positive real values with
fixed, then
10.19.3
10.19.4
If
through positive real values with
fixed, and
10.19.5
then
10.19.6
10.19.7
In these expansions
and
are the polynomials in
of degree
defined in §10.41(ii).
§10.19(iii) Transition Region
As
, with
fixed,
10.19.8
,
.
Also,
10.19.9
10.19.12
,
.
10.19.13
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).

