# §6.12 Asymptotic Expansions

## §6.12(i) Exponential and Logarithmic Integrals

When the remainder is bounded in magnitude by the first neglected term, and has the same sign when . When the remainder term is bounded in magnitude by times the first neglected term. For these and other error bounds see Olver (1997b, pp. 109–112) with .

For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)2.11(iv), with .

If the expansion is terminated at the th term, then the remainder term is bounded by times the next term. For the function see §9.7(i).

The asymptotic expansion of as is obtainable from (6.2.8) and (6.12.2).

## §6.12(ii) Sine and Cosine Integrals

The asymptotic expansions of and are given by (6.2.19), (6.2.20), together with

as in .

The remainder terms are given by

where, for ,

When , these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when . When the remainders are bounded in magnitude by times the first neglected terms.

For other phase ranges use (6.4.6) and (6.4.7). For exponentially-improved asymptotic expansions, use (6.5.5), (6.5.6), and §6.12(i).