On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order

C. J. Howls, A. B.Olde Daalhuis

Abstract

For the coefficients An(ζ) and Bn(ζ), that occur in the uniform asymptotic expansions of Bessel functions of large order, we give asymptotic expansions as n →∞. The coefficients in these asymptotic expansions are again Am>(ζ) and Bm(ζ), and the asymptotic base consists of functions Apq(n,ζ), which can be seen as new generalizations of the Airy function. We describe the asymptotic behaviour of the functions Apq(n, ζ), as n→∞, and we compute the Taylor–series expansions of Apq(n, ζ) at ζ = 0. Two numerical examples are included.