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The generalized exponential integral. (English) Zbl 0821.33001
Zahar, R. V. M. (ed.), Approximation and computation: a Festschrift in honor of Walter Gautschi. Proceedings of the Purdue conference, West Lafayette, IN, USA, December 2-5, 1993. Boston, US: Birkhäuser. ISNM, Int. Ser. Numer. Math. 119, 497-510 (1994).
The paper discusses the role of the generalized exponential integral in recently-developed theories of exponentially-improved asymptotic expansions and the Stokes phenomenon. A uniform asymptotic behavior of the integral is given when both (complex) parameters are large in absolute value and the remainder is re-expanded in order to get an exponentially-improved expansion. Applications to other functions are given by considering a class of linear ordinary differential equations of second order.
MSC:
33B20Incomplete beta and gamma functions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)