# §13.20 Uniform Asymptotic Approximations for Large

## §13.20(i) Large , Fixed

When in the sector , with fixed

uniformly for bounded values of ; also

uniformly for bounded positive values of . For an extension of (13.20.1) to an asymptotic expansion, together with error bounds, see Olver (1997b, Chapter 10, Ex. 3.4).

## §13.20(ii) Large ,

Let

13.20.3

Then as

uniformly with respect to and , where again denotes an arbitrary small positive constant.

## §13.20(iii) Large ,

Let

13.20.6
13.20.7
13.20.8

with the variable defined implicitly as follows:

(b) In the case

the upper or lower sign being taken according as .

(In both cases (a) and (b) the -interval is mapped one-to-one onto the -interval , with and corresponding to and , respectively.) Then as

uniformly with respect to and . For the parabolic cylinder function see §12.2.

These results are proved in Olver (1980b). This reference also supplies error bounds and corresponding approximations when , , and are replaced by , , and , respectively.

## §13.20(iv) Large ,

Again define , , and by (13.20.6)–(13.20.8), but with now defined by

uniformly with respect to and .

For the parabolic cylinder functions and see §12.2, and for the functions associated with and see §14.15(v).

These results are proved in Olver (1980b). Equations (13.20.17) and (13.20.18) are simpler than (6.10) and (6.11) in this reference. Olver (1980b) also supplies error bounds and corresponding approximations when , , and are replaced by , , and , respectively.

It should be noted that (13.20.11), (13.20.16), and (13.20.18) differ only in the common error terms. Hence without the error terms the approximation holds for . Similarly for (13.20.12), (13.20.17), and (13.20.19).

## §13.20(v) Large , Other Expansions

For uniform approximations valid when is large, , and , see Olver (1997b, pp. 401–403). These approximations are in terms of Airy functions.

For uniform approximations of and , and real, one or both large, see Dunster (2003a).