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Zbl 0973.33002
Howls, C.J.; Olde Daalhuis, A.B.
On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No.1991, 3917-3930 (1999). ISSN 1364-5021; ISSN 1471-2946/e

If the coefficients $a_n$ in an asymptotic expansion are related to themselves or to coefficients $a_m$ for $n\to\infty$, the expansion is said to have the resurgence property. This property has been shown to be true for many Poincaré expansions, cf. {\it M. V. Berry} and {\it C. J. Howls} [Proc. R. Soc. Lond., Ser. A 434, 657-675 (1991; Zbl 0764.30031] or {\it C. J. Howls} [Proc. R. Soc. Lond., Ser. A 453, 2271-2294 (1997; Zbl 1067.58501)]. The authors show the resurgence property for the classical uniform asymptotic expansion for the Bessel functions $J_\nu(\nu z)$. They recall two different approaches which lead to this expansion. First the differential equation point of view and second an integral representation using Bleistein's method, cf. {\it N. Bleistein} [Commun. Pure Appl. Math. 19, 353-370 (1966; Zbl 0145.15801)]. Again by Bleistein's method the authors then derive a new expansion of the coefficients which leads to the desired asymptotics for $n\to\infty$.
[Wolfgang Castell (Erlangen)]

MSC 2000:: *33C10 Cylinder functions, etc.
34E05 Asymptotic expansions (ODE)
34E20 Asymptotic singular perturbations, methods (ODE)

Keywords: Airy function; asymptotic expansions; Bessel functions; steepest descent; turning points

Citations: Zbl 0794.30031; Zbl 0145.15801; Zbl 1067.58501; Zbl 0764.30031

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