§7.12 Asymptotic Expansions

As $z\to \mathrm{\infty }$

both expansions being valid when $|\mathrm{ph}z|\le \frac{3}{4}\pi -\delta$ ().

When $|\mathrm{ph}z|\le \frac{1}{4}\pi$ the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when $\mathrm{ph}z=0$. When the remainder terms are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ times the first neglected terms. For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by $z^2$; compare (7.11.2).

For re-expansions of the remainder terms leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv) and use (7.11.3). (Note that some of these re-expansions themselves involve the complementary error function.)

The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and

as $z\to \mathrm{\infty }$ in . The remainder terms are given by

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

When $|\mathrm{ph}z|\le \frac{1}{8}\pi$, $R_n^{(\mathrm{f})}(z)$ and $R_n^{(\mathrm{g})}(z)$ are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. They are bounded by $|\csc \left(4\mathrm{ph}z\right)|$ times the first neglected terms when .

For other phase ranges use (7.4.7) and (7.4.8). For exponentially-improved expansions use (7.5.7), (7.5.10), and §7.12(i).

See Olver (1997b, p. 115) for an expansion of $G\left(z\right)$ with bounds for the remainder for real and complex values of $z$.