All of (11.11.10)–(11.11.17) can be regarded as special
cases of two asymptotic expansions given in
Olver (1997b, pp. 352–357) for as
, one being uniform for , where
again denotes an arbitrary small positive constant, and the other
being uniform for . (Note that Olver’s definition of
omits the factor in (11.10.4).) See also
Watson (1944, §10.15).
Lastly, corresponding asymptotic approximations and expansions for
and follow from
(11.10.15) and (11.10.16) and the corresponding asymptotic
expansions for the Bessel functions and
; see §10.19(ii). In particular,