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On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. (English) Zbl 0973.33002
If the coefficients a n in an asymptotic expansion are related to themselves or to coefficients a m for n, the expansion is said to have the resurgence property. This property has been shown to be true for many Poincaré expansions, cf. M. V. Berry and C. J. Howls [Proc. R. Soc. Lond., Ser. A 434, 657-675 (1991; Zbl 0764.30031] or 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 ν (νz). 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. 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.
33C10Bessel and Airy functions, cylinder functions, 0 F 1
34E05Asymptotic expansions (ODE)
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)