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11: 9.7 Asymptotic Expansions

… ►In (9.7.5) and (9.7.6) the $n$th error term, that is, the error on truncating the expansion at $n$ 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 $n\ge 0$ for (9.7.5) and $n\ge 1$ for (9.7.6). ►In (9.7.7) and (9.7.8) 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. ►In (9.7.9)–(9.7.12) the $n$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. … ►The $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by …

12: Preface

… ►Notwithstanding the great care that has been exercised by the editors, authors, validators, and the NIST staff, it is almost inevitable that in a work of the magnitude and scope of the NIST Handbook and DLMF errors will still be present. …

13: 8.7 Series Expansions

… ►For an expansion for $\gamma (a,\mathrm{i}x)$ in series of Bessel functions $J_n\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\ge 0$) is small or moderate in magnitude see Barakat (1961).

14: 1.6 Vectors and Vector-Valued Functions

… ►

Magnitude and Angle of Vector $𝐚$

►

1.6.3 $$\Vert 𝐚\Vert =\sqrt{𝐚\cdot 𝐚},$$

ⓘ

Symbols:

$\Vert 𝐯\Vert$: vector norm ($l$²)

Permalink:

http://dlmf.nist.gov/1.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §1.6(i), §1.6(i), §1.6 and Ch.1

…

15: 10.74 Methods of Computation

… ►As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. …

16: 30.16 Methods of Computation

… ►and real eigenvalues ${\alpha }_{1,d}$, ${\alpha }_{2,d}$, $\mathrm{\dots }$, ${\alpha }_{d,d}$, arranged in ascending order of magnitude. …

17: 3.6 Linear Difference Equations

… ►This is of little consequence if the wanted solution is growing in magnitude at least as fast as any other solution of (3.6.3), and the recursion process is stable. … ►If, as $n\to \mathrm{\infty }$, the wanted solution $w_n$ grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable. …

18: 5.11 Asymptotic Expansions

… ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ($k\ge 0$) and $z$ is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. If $z$ is complex, then the remainder terms are bounded in magnitude by ${\sec }^{2n}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.1), and ${\sec }^{2n+1}\left(\frac{1}{2}\mathrm{ph}z\right)$ for (5.11.2), times the first neglected terms. …

19: 13.9 Zeros

… ►When and $b>0$ let ${\varphi }_r$, $r=1,2,3,\mathrm{\dots }$, be the positive zeros of $M(a,b,x)$ arranged in increasing order of magnitude, and let $j_{b-1,r}$ be the $r$th positive zero of the Bessel function $J_{b-1}\left(x\right)$ (§10.21(i)). …

20: 19.27 Asymptotic Approximations and Expansions

… ►A similar (but more general) situation prevails for $R_{-a}(𝐛;𝐳)$ when some of the variables $z_1,\mathrm{\dots },z_n$ are smaller in magnitude than the rest; see Carlson (1985, (4.16)–(4.19) and (2.26)–(2.29)). …