§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
For asymptotic expansions of
as
in various
sectors of the complex
-plane for fixed real values of
and fixed
real or complex values of
, see Wright (1935) when
,
and Wright (1940b) when
. For exponentially-improved
asymptotic expansions in the same circumstances, together with smooth
interpretations of the corresponding Stokes phenomenon
(§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b)
when
, and Wong and Zhao (1999a) when
.
The Laplace transform of
can be expressed in terms of the
Mittag-Leffler function:
See Paris (2002c). This reference includes exponentially-improved
asymptotic expansions for
when
, together with a
smooth interpretation of Stokes phenomena. See also Wong and Zhao (2002a),
and for further information on the Mittag-Leffler function see
Erdélyi et al. (1955, §18.1) and Paris and Kaminski (2001, §5.1.4).
For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).


