§9.7 Asymptotic Expansions
Contents
- §9.7(i) Notation
- §9.7(ii) Poincaré-Type Expansions
- §9.7(iii) Error Bounds for Real Variables
- §9.7(iv) Error Bounds for Complex Variables
- §9.7(v) Exponentially-Improved Expansions
§9.7(i) Notation
Here denotes an arbitrary small positive constant and
Also and for
Lastly,
Numerical values of this function are given in Table 9.7.1 for to 2D. For large ,
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(ii) Poincaré-Type Expansions
As the following asymptotic expansions are valid uniformly in the stated sectors.
§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.
As special cases, when
where .
§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
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
with . Then
where
(For the notation see §8.2(i).) And as with fixed