Asymptotic expansions for Riesz potentials of Airy functions and their products

Abstract

Abstract. Riesz potentials of a function are defined as fractional powers of the Laplacian. Asymptotic expansions for x → ±∞ are derived for the Riesz potentials of the Airy function Ai(x) and the Scorer function Gi(x). Reduction formulas are provided that allow to compute Riesz potentials of the products of Airy functions Ai 2 (x) and Ai(x)Bi(x), where Bi(x) is the Airy function of the second type, via the Riesz potentials of Ai(x) and Gi(x). Integral representations are given for the function A 2 (a, b; x) = Ai (x − a) Ai (x − b) with a, b ∈ R, and its Hilbert transform. Combined with the above asymptotic expansions they can be used for obtaining asymptotics of the Hankel transform of Riesz potentials of A 2 (a, b; x). The study of the above Riesz fractional derivatives can be used for establishing new properties of Korteweg-de Vries-type equations.

Riesz potentials for Korteweg-de Vries solitons

“…The upper bound on α can be eliminated by modifying the contour of integration in (2.3) in the complex plain. For the Airy functions it should go into the valley of the exponential exp(iξ) 3 (see [19]). …”