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10 Bessel FunctionsModified Bessel Functions

§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

The function ϕ(ρ,β;z) is defined by

10.46.1 ϕ(ρ,β;z)=k=0zkk!Γ(ρk+β),

From (10.25.2)

10.46.2 Iν(z)=(12z)νϕ(1,ν+1;14z2).

For asymptotic expansions of ϕ(ρ,β;z) as z in various sectors of the complex z-plane for fixed real values of ρ and fixed real or complex values of β, see Wright (1935) when ρ>0, and Wright (1940b) when -1<ρ<0. 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 ρ>0, and Wong and Zhao (1999a) when -1<ρ<0.

The Laplace transform of ϕ(ρ,β;z) can be expressed in terms of the Mittag-Leffler function:

10.46.3 Ea,b(z)=k=0zkΓ(ak+b),

See Paris (2002c). This reference includes exponentially-improved asymptotic expansions for Ea,b(z) when |z|, 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), Paris and Kaminski (2001, §5.1.4), and Haubold et al. (2011).

For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).