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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. (English) Zbl 1152.34394
Summary: Second-order linear ordinary differential equations with a large parameter $u$ are examined. Classic asymptotic expansions involving Airy functions are applicable for the case where the argument $z$ lies in complex domain containing a simple turning point. In this article, such asymptotic expansions are converted into convergent series, where $u$ appears in an inverse factorial, rather than an inverse power. The domain of convergence of the new expansions is rigorously established and is found to be an unbounded domain containing the turning point. The theory is then applied to obtain convergent expansions for Bessel functions of complex argument and large positive order.
MSC:
 34M30 Asymptotics, summation methods (ODE in the complex domain) 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$