# §8.11 Asymptotic Approximations and Expansions

## §8.11(i) Large , Fixed

If is real and () is positive, then is bounded in absolute value by the first neglected term and has the same sign provided that . For bounds on when is real and is complex see Olver (1997b, pp. 109–112). For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a).

## §8.11(ii) Large , Fixed

This expansion is absolutely convergent for all finite , and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of as in .

## §8.11(iii) Large , Fixed

The expansion (8.11.7) also applies when with , and in this case Gautschi (1959a) supplies numerical bounds for the remainders in the truncated expansion (8.11.7). For extensions to complex variables see Temme (1994b, §4), and also Mahler (1930), Tricomi (1950b), and Paris (2002b).

## §8.11(iv) Large , Bounded

in both cases uniformly with respect to bounded real values of . For Dawson’s integral see §7.2(ii). See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. For related expansions involving Hermite polynomials see Pagurova (1965).

## §8.11(v) Other Approximations

For the function defined by (8.4.11),

8.11.13

With , an asymptotic expansion of follows from (8.11.14) and (8.11.16).

For (8.11.18) and extensions to complex values of see Buckholtz (1963). For a uniformly valid expansion for and , see Wong (1973b).