# §13.7 Asymptotic Expansions for Large Argument

## §13.7(ii) Error Bounds

Figure 13.7.1: Regions , , , , and are the closures of the indicated unshaded regions bounded by the straight lines and circular arcs centered at the origin, with .

where

and with the notation of Figure 13.7.1

13.7.6
,
,
,

according as

13.7.7
,
,
,

respectively, with

13.7.8
,
,
.

Also, when

13.7.9
,
,
,

and when is replaced by and is replaced by everywhere in (13.7.9).

For numerical values of see Table 9.7.1.

Corresponding error bounds for (13.7.2) can be constructed by combining (13.2.41) with (13.7.4)–(13.7.9).

## §13.7(iii) Exponentially-Improved Expansion

For proofs see Olver (1991b, 1993a). For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).