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Zbl 1039.33500
Paris, R. B.
On the use of Hadamard expansions in hyperasymptotic evaluation. I: Real variables. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457, No. 2016, 2835-2853 (2001). ISSN 1364-5021; ISSN 1471-2946/e

It is known that the level of accuracy afforded by optimal truncation of an asymptotic expansion (superasymptotic level) results in a remainder term that is exponentially small in the asymptotic variable. In order to achieve higher levels of accuracy a hyperasymptotic expansion method was introduced by {\it M. V. Berry} and {\it C. J. Howls} [Proc. R. Soc. Lond., Ser. A 434, No. 1892, 657--675 (1991; Zbl 0764.30031); ibid. 430, No. 1880, 653--668 (1990; Zbl 0745.34052)] by re-expanding the superasymptotic remainder in a new asymptotic series whose remainder is exponentially small compared with the first remainder. This process is repeated by expanding the remainder in a new expansion up to any desired level of accuracy. In this paper the author proposes a new method of hyperasymptotic evaluation using (absolutely convergent) Hadamard expansions. The essence of the method is previously illustrated by evaluating a slowly convergent Hadamard expansion of a well-known integral representation of the modified Bessel function $I_\nu(x)$ (for $x>0)$ written in terms of the incomplete gamma function [see {\it G. N. Watson}, A treatise on the theory of Bessel functions, 2nd. ed., Cambridge Math. Library (ed. 1995; Zbl 0849.33001)]. This expansion is here transformed by a simple rearrangement of the tail to produce a rapidly convergent series which is suitable for hyperasymptotic levels of precision. Then, starting from an integral representation for the confluent hypergeometric function $_1F_1 (a;b;x)$ (when $x>0)$ [see {\it M. Abramowitz} and {\it I. Stegun} (eds.), Handbook of mathematical functions, 10 th. printing, New-York: Wiley, p. 505 (1972; Zbl 0543.33001)] (function that includes $I_\nu(x)$ as a particular typical case), the author derives a modified Hadamard expansion that consists of a single sum with coefficients involving the normalized incomplete gamma function $P(b-a+k, x)$ Re\,$b>\text{Re}\,a>0)$. When $a=\nu+\frac 12$, $b=2\nu+1$ the results agree with those obtained for $I_\nu(x)$. By an extension of the previous analysis, the Hadamard expansion for the second type of confluent hypergeometric function $U(a;b;x)$ $(x>0)$, represented by a Laplace integral over an infinite interval [Abramowitz and Stegun, ibid, p. 505] it requires a decomposition of the path of integration to yield an infinite number of Hadamard expansions, associated with a decreasing sequence of subdominant exponentials. Numerical examples involving $I_\nu(x)$ and $K_\nu(x)$ are also given to illustrate the accuracy of the asymptotic expansions.
[Nácere Hayek (La Laguna)]

MSC 2000:: *33C15 Confluent hypergeometric functions
33F05 Numerical approximation of special functions
65D20 Computation of special functions

Citations: Zbl 0764.30031; Zbl 0745.34052; Zbl 0849.33001; Zbl 0543.33001

Cited in: Zbl 1097.33503

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