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21: 4.37 Inverse Hyperbolic Functions

… ►the upper/lower signs corresponding to the right/left sides. … ►the upper or lower sign being taken according as

\Re z\gtrless 0

; compare Figure 4.37.1(ii). … ►the upper/lower sign corresponding to the upper/lower side. … ►the upper/lower sign corresponding to the upper/lower side. … ►the upper/lower sign corresponding to the upper/lower sides. …

22: 19.34 Mutual Inductance of Coaxial Circles

… ►is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …

23: 34.1 Special Notation

… ►

34.1.1

\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};

ⓘ

Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Sources:: Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))
Permalink:: http://dlmf.nist.gov/34.1.E1
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.15):: A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3j}$ . The sign of $m_{3}$ was changed for clarity.
See also:: Annotations for §34.1 and Ch.34

…

24: 3.8 Nonlinear Equations

… ►

(a)

$f(x_{0})f^{\prime\prime}(x_{0})>0$ and $f^{\prime}(x)$ , $f^{\prime\prime}(x)$ do not change sign between $x_{0}$ and $\xi$ (monotonic convergence).

►

(b)

$f(x_{0})f^{\prime\prime}(x_{0})<0$ , $f^{\prime}(x)$ , $f^{\prime\prime}(x)$ do not change sign in the interval $(x_{0},x_{1})$ , and $\xi\in[x_{0},x_{1}]$ (monotonic convergence after the first iteration).

… ►Let

x_{0}

and

x_{1}

be such that

f_{0}=f(x_{0})

and

f_{1}=f(x_{1})

have opposite signs. …We continue with

x_{2}

and either

x_{0}

x_{1}

, depending which of

f_{0}

and

f_{1}

is of opposite sign to

f(x_{2})

, and so on. … ►Whether or not

f_{0}

and

f_{1}

have opposite signs,

x_{2}

is computed as in (3.8.6). …

25: 4.24 Inverse Trigonometric Functions: Further Properties

… ►

4.24.4

\operatorname{arctan}z=\pm\frac{\pi}{2}-\frac{1}{z}+\frac{1}{3z^{3}}-\frac{1}{% 5z^{5}}+\cdots,

\Re z\gtrless 0

|z|\geq 1

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $z$ : complex variable
A&S Ref:: 4.4.42 (has an error. $\frac{\pi}{2}$ should have a negative sign when $\Re z<0$ .)
Referenced by:: §4.45(i)
Permalink:: http://dlmf.nist.gov/4.24.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

…

26: 6.12 Asymptotic Expansions

… ►When

|\operatorname{ph}z|\leq\frac{1}{2}\pi

the remainder is bounded in magnitude by the first neglected term, and has the same sign when

\operatorname{ph}z=0

. … ►When

|\operatorname{ph}z|\leq\tfrac{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