The Generalized Exponential Integral
This paper concerns the role of the generalized exponential integral in recently-developed theories of exponentially-improved asymptotic expansions and the Stokes phenomenon. The first part describes the asymptotic behavior of the integral when both the argument and order are large in absolute value. The second part shows how to increase the accuracy of asymptotic expansions of solutions of linear differential equations of the second order by re-expanding the remainder terms in series of generalized exponential integrals.
KeywordsAsymptotic Expansion Linear Differential Equation Remainder Term Small Term Incomplete Gamma Function
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- Airey J. R. The “converging factor” in asymptotic series and the calculation of Bessel, Laguerre and other functions. Philos. Mag. , 24: 521–552, 1937.Google Scholar
- Olver F. W. J. Asymptotics and Special Functions. Academic Press, New York, 1974.Google Scholar
- Olver F. W. J. On Stokes’ phenomenon and converging factors. In Asymptotic and Computational Analysis, Lecture Notes in Pure and Applied Mathematics, 124:329–355. Marcel Dekker, New York, 1990. R. Wong, ed.Google Scholar
- Olver F. W. J. Converging factors. In Wave Asymptotics, pages 54–68. Cambridge University Press, 1992. P. A. Martin and G. R. Wickham, eds.Google Scholar
- Stieltjes T. J. Recherches sur quelques séries semi-convergentes. Ann. Sci. École Norm. Sup. , 3:201–258, 1886. Reprinted in Complete Works. Noordhoff, Groningen, 2: 2–58, 1918.Google Scholar