# §9.7 Asymptotic Expansions

## §9.7(i) Notation

Here denotes an arbitrary small positive constant and

9.7.1

Also and for

Lastly,

9.7.3

Numerical values of this function are given in Table 9.7.1 for to 2D. For large ,

9.7.4
Table 9.7.1: .
1 1.57 6 3.20 11 4.25 16 5.09
2 2.00 7 3.44 12 4.43 17 5.24
3 2.36 8 3.66 13 4.61 18 5.39
4 2.67 9 3.87 14 4.77 19 5.54
5 2.95 10 4.06 15 4.94 20 5.68

## §9.7(iii) Error Bounds for Real Variables

In (9.7.5) and (9.7.6) the th error term, that is, the error on truncating the expansion at terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if for (9.7.5) and for (9.7.6).

In (9.7.7) and (9.7.8) the th error term is bounded in magnitude by the first neglected term multiplied by where for (9.7.7) and for (9.7.8), provided that in both cases.

In (9.7.9)–(9.7.12) the th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign.

## §9.7(iv) Error Bounds for Complex Variables

When the th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

according as , , or . Here for (9.7.5) and for (9.7.6).

Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms.

For other error bounds see Boyd (1993).

## §9.7(v) Exponentially-Improved Expansions

In (9.7.5) and (9.7.6) let

with . Then

where

(For the notation see §8.2(i).) And as with fixed

For re-expansions of the remainder terms in (9.7.7)–(9.7.14) combine the results of this section with those of §9.2(v) and their differentiated forms, as in §9.7(iv).

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