# §13.8 Asymptotic Approximations for Large Parameters

## §13.8(i) Large , Fixed and

When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when is large, and and are bounded.

## §13.8(ii) Large and , Fixed and

Let and with . Then

and

as , uniformly in compact -intervals of and compact real -intervals. For the parabolic cylinder function see §12.2, and for an extension to an asymptotic expansion see Temme (1978).

To obtain approximations for and that hold as , with and combine (13.14.4), (13.14.5) with §13.20(i).

Also, more complicated—but more powerful—uniform asymptotic approximations can be obtained by combining (13.14.4), (13.14.5) with §§13.20(iii) and 13.20(iv).

For other asymptotic expansions for large and see López and Pagola (2010).

## §13.8(iii) Large

For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv).

When with () fixed,

where , and . (13.8.8) holds uniformly with respect to . For the case the transformation (13.2.40) can be used.

For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i).

For asymptotic approximations to and as that hold uniformly with respect to and bounded positive values of , combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii).