# §10.40 Asymptotic Expansions for Large Argument

## §10.40(i) Hankel’s Expansions

With the notation of §§10.17(i) and 10.17(ii), as with fixed,

Corresponding expansions for , , , and for other ranges of are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). In particular, use of (10.34.3) with yields the following more general (and more accurate) version of (10.40.1):

### ¶ Products

With and fixed,

as in . The general terms in (10.40.6) and (10.40.7) can be written down by analogy with (10.18.17), (10.18.19), and (10.18.20).

## §10.40(ii) Error Bounds for Real Argument and Order

In the expansion (10.40.2) assume that and the sum is truncated when . Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that .

For the error term in (10.40.1) see §10.40(iii).

## §10.40(iii) Error Bounds for Complex Argument and Order

For (10.40.2) write

Then

where denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that changes monotonically. Bounds for are given by

where ; see §9.7(i).

A similar result for (10.40.1) is obtained by combining (10.34.3), with , and (10.40.10)–(10.40.12); see Olver (1997b, p. 269).

## §10.40(iv) Exponentially-Improved Expansions

For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).