# §13.21 Uniform Asymptotic Approximations for Large

## §13.21(i) Large , Fixed

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

Other types of approximations when through positive real values with () fixed are as follows. Define

Then

uniformly with respect to .

For (13.21.6), (13.21.7), and extensions to asymptotic expansions and error bounds, see Olver (1997b, Chapter 12, Exs. 12.4.5, 12.4.6). For extensions to complex values of see López (1999).

## §13.21(ii) Large ,

Let

13.21.8
13.21.9
13.21.10

with the variable defined implicitly by

13.21.15
13.21.16

uniformly with respect to and , where again denotes an arbitrary small positive constant. For the functions , , , and see §10.2(ii), and for the functions associated with and see §2.8(iv).

These approximations are proved in Dunster (1989). This reference also includes error bounds and extensions to asymptotic expansions and complex values of .

## §13.21(iii) Large , (Continued)

Let

13.21.17
13.21.18
13.21.19

and define the variable implicitly by

uniformly with respect to and . For the functions and see §9.2(i), and for the functions associated with and see §2.8(iii).

These approximations are proved in Dunster (1989). This reference also includes error bounds and extensions to asymptotic expansions and complex values of .

## §13.21(iv) Large , Other Expansions

For a uniform asymptotic expansion in terms of Airy functions for when is large and positive, is real with bounded, and see Olver (1997b, Chapter 11, Ex. 7.3). This expansion is simpler in form than the expansions of Dunster (1989) that correspond to the approximations given in §13.21(iii), but the conditions on are more restrictive.

For asymptotic expansions having double asymptotic properties see Skovgaard (1966).