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On the use of Hadamard expansions in hyperasymptotic evaluation. I: Real variables. (English) Zbl 1039.33500
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 M. V. Berry and 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 ${I}_{\nu }\left(x\right)$ (for $x>0\right)$ written in terms of the incomplete gamma function [see 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 ${}_{1}{F}_{1}\left(a;b;x\right)$ (when $x>0\right)$ [see M. Abramowitz and I. Stegun (eds.), Handbook of mathematical functions, 10 th. printing, New-York: Wiley, p. 505 (1972; Zbl 0543.33001)] (function that includes ${I}_{\nu }\left(x\right)$ 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 $P\left(b-a+k,x\right)$ Re $b>\text{Re}\phantom{\rule{0.166667em}{0ex}}a>0\right)$. When $a=\nu +\frac{1}{2}$, $b=2\nu +1$ the results agree with those obtained for ${I}_{\nu }\left(x\right)$. By an extension of the previous analysis, the Hadamard expansion for the second type of confluent hypergeometric function $U\left(a;b;x\right)$ $\left(x>0\right)$, 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 ${I}_{\nu }\left(x\right)$ and ${K}_{\nu }\left(x\right)$ are also given to illustrate the accuracy of the asymptotic expansions.

##### MSC:
 33C15 Confluent hypergeometric functions, Whittaker functions, ${}_{1}{F}_{1}$ 33F05 Numerical approximation and evaluation of special functions 65D20 Computation of special functions, construction of tables