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

$x>0)$ 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

${}_{1}{F}_{1}(a;b;x)$ (when

$x>0)$ [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

$P(b-a+k,x)$ Re

$b>\text{Re}\phantom{\rule{0.166667em}{0ex}}a>0)$. 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

$U(a;b;x)$ $(x>0)$, 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

${I}_{\nu}\left(x\right)$ and

${K}_{\nu}\left(x\right)$ are also given to illustrate the accuracy of the asymptotic expansions.