error term

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21: 2.8 Differential Equations with a Parameter

… ►An alternative way of representing the error terms in (2.8.15) and (2.8.16) is as follows. … ►Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions. …

22: 2.11 Remainder Terms; Stokes Phenomenon

… ►The error term is, in fact, approximately 700 times the last term obtained in (2.11.4). … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). …Hence from §7.12(i) $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. … ►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. …

23: Errata

… ►

Bounds have been sharpened. The second paragraph now reads, “The $n$th error term is bounded in magnitude by the first neglected term multiplied by $\chi (n+\sigma )+1$ where $\sigma =\frac{1}{6}$ for (9.7.7) and $\sigma =0$ for (9.7.8), provided that $n\ge 0$ in the first case and $n\ge 1$ in the second case.” Previously it read, “In (9.7.7) and (9.7.8) the $n$th error term is bounded in magnitude by the first neglected term multiplied by $2\chi (n)\exp \left(\sigma \pi /(72\zeta )\right)$ where $\sigma =5$ for (9.7.7) and $\sigma =7$ for (9.7.8), provided that $n\ge 1$ in both cases.” In Equation (9.7.16)

9.7.16

$$\mathrm{Bi}\left(x\right)\le \frac{{\mathrm{e}}^{\xi }}{\sqrt{\pi }{x}^{1/4}}\left(1+\left(\chi (\frac{7}{6})+1\right)\frac{5}{72\xi }\right),$$

$${\mathrm{Bi}}^{\prime }\left(x\right)\le \frac{x^{1/4}{\mathrm{e}}^{\xi }}{\sqrt{\pi }}\left(1+\left(\frac{\pi }{2}+1\right)\frac{7}{72\xi }\right),$$

the bounds on the right-hand sides have been sharpened. The factors $\left(\chi (\frac{7}{6})+1\right)\frac{5}{72\xi }$, $\left(\frac{\pi }{2}+1\right)\frac{7}{72\xi }$, were originally given by $\frac{5\pi }{72\xi }\exp \left(\frac{5\pi }{72\xi }\right)$, $\frac{7\pi }{72\xi }\exp \left(\frac{7\pi }{72\xi }\right)$, respectively.

►

Bounds have been sharpened. The first paragraph now reads, “The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17

provided that $n\ge 0$, $\sigma =\frac{1}{6}$ for (9.7.5) and $n\ge 1$, $\sigma =0$ for (9.7.6).” Previously it read, “When $n\ge 1$ the $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

9.7.17

Here $\sigma =5$ for (9.7.5) and $\sigma =7$ for (9.7.6).”

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24: 2.5 Mellin Transform Methods

… ►The first reference also contains explicit expressions for the error terms, as do Soni (1980) and Carlson and Gustafson (1985). …

25: 10.21 Zeros

… ►(Note: If the term $z(\zeta ){(h(\zeta ))}^2{C}_0(\zeta )/(2\zeta \nu )$ in (10.21.43) is omitted, then the uniform character of the error term $O(1/\nu )$ is destroyed.) …

26: 7.7 Integral Representations

… ►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. …

27: 6.12 Asymptotic Expansions

… ►For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$. …

28: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). …

29: 7.12 Asymptotic Expansions

… ►

§7.12(i) Complementary Error Function

… ►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 $\mathrm{ph}z=0$. …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). … ►(Note that some of these re-expansions themselves involve the complementary error function.) … ►The remainder terms are given by …

30: 19.27 Asymptotic Approximations and Expansions

… ►Although they are obtained (with some exceptions) by approximating uniformly the integrand of each elliptic integral, some occur also as the leading terms of known asymptotic series with error bounds (Wong (1983, §4), Carlson and Gustafson (1985), López (2000, 2001)). …