§6.12 Asymptotic Expansions

When $|\mathrm{ph}z|\le \frac{1}{2}\pi$ the remainder is bounded in magnitude by the first neglected term, and has the same sign when $\mathrm{ph}z=0$. When the remainder term is bounded in magnitude by $\csc \left(|\mathrm{ph}z|\right)$ times the first neglected term. For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$.

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 $p=1$.

If the expansion is terminated at the $n$th term, then the remainder term is bounded by $1+\chi (n+1)$ times the next term. For the function $\chi$ see §9.7(i).

The asymptotic expansion of $\mathrm{li}\left(x\right)$ as $x\to \mathrm{\infty }$ is obtainable from (6.2.8) and (6.12.2).

The asymptotic expansions of $\mathrm{Si}\left(z\right)$ and $\mathrm{Ci}\left(z\right)$ are given by (6.2.19), (6.2.20), together with

as $z\to \mathrm{\infty }$ in .

The remainder terms are given by

where, for $n=0,1,2,\mathrm{\dots }$,

When $|\mathrm{ph}z|\le \frac{1}{4}\pi$, 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 $\mathrm{ph}z=0$. When the remainders are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ 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).