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1: 26.13 Permutations: Cycle Notation

… ►is $(1,3,2,5,7)(4)(6,8)$ in cycle notation. …In consequence, (26.13.2) can also be written as $(1,3,2,5,7)(6,8)$. … ►For the example (26.13.2), this decomposition is given by $(1,3,2,5,7)(6,8)=(1,3)(2,3)(2,5)(5,7)(6,8).$ … ►If , then $(j,k)$ is a product of $2k-2j-1$ adjacent transpositions: …Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $(1,3,2,5,7)(6,8)=(2,3)(1,2)(4,5)(3,4)(2,3)(3,4)(4,5)(6,7)(5,6)(7,8)(6,7)$: $inv((1,3,2,5,7)(6,8))=11$.

2: 7.23 Tables

… ►

Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 6S; $F\left(x\right)$, $x\in [0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in [0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in [0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in [0,1]$, $x^{-1}\in [0,1]$, 15D.

… ►

Finn and Mugglestone (1965) includes the Voigt function $H(a,u)$, $u\in [0,22]$, $a\in [0,1]$, 6S.

… ►

Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

…

3: 14.27 Zeros

… ► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ has no zeros in the interval $(-\mathrm{\infty },-1)$. …

4: 26.15 Permutations: Matrix Notation

… ►If $(j,k)\in B$, then $\sigma (j)\ne k$. The number of derangements of $n$ is the number of permutations with forbidden positions $B=\{(1,1),(2,2),\mathrm{\dots },(n,n)\}$. … ►For $(j,k)\in B$, $B\setminus [j,k]$ denotes $B$ after removal of all elements of the form $(j,t)$ or $(t,k)$, $t=1,2,\mathrm{\dots },n$. $B\setminus (j,k)$ denotes $B$ with the element $(j,k)$ removed. … ►Let . …

5: 14.16 Zeros

… ►

§14.16(ii) Interval

… ►The zeros of ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ interlace those of ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$. … ►

§14.16(iii) Interval

► $P_{\nu }^{\mu }\left(x\right)$ has exactly one zero in the interval $(1,\mathrm{\infty })$ if either of the following sets of conditions holds: … ► ${\mathit{Q}}_{\nu }^{\mu }\left(x\right)$ has no zeros in the interval $(1,\mathrm{\infty })$ when $\nu >-1$, and at most one zero in the interval $(1,\mathrm{\infty })$ when .

6: 3.7 Ordinary Differential Equations

… ►where $\mathbf{A}(\tau ,z)$ is the matrix ►

3.7.6

ⓘ

Defines:

$\mathbf{A}$: matrix (locally) and $\mathbf{A}$: matrix (locally)

Symbols:

$(a,b)$: open interval

Permalink:

http://dlmf.nist.gov/3.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(ii), §3.7 and Ch.3

►and $\mathbf{b}(\tau ,z)$ is the vector … ►Let $(a,b)$ be a finite or infinite interval and $q(x)$ be a real-valued continuous (or piecewise continuous) function on the closure of $(a,b)$. … ►If $q(x)$ is $C^{\mathrm{\infty }}$ on the closure of $(a,b)$, then the discretized form (3.7.13) of the differential equation can be used. …

7: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

… ►Jacobian elliptic functions with real moduli in the intervals $(-\mathrm{\infty },0)$ and $(1,\mathrm{\infty })$, or with purely imaginary moduli are related to functions with moduli in the interval $[0,1]$ by the following formulas. … ►For proofs of these results and further information see Walker (2003).

8: 1.4 Calculus of One Variable

… ► … ►Suppose $f(x)$ is defined on $[a,b]$. … ►A function $f(x)$ is convex on $(a,b)$ if …for any $c,d\in (a,b)$, and $t\in [0,1]$. …A similar definition applies to closed intervals $[a,b]$. …

9: 32.4 Isomonodromy Problems

… ►

32.4.4

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Symbols:

$\mathrm{i}$: imaginary unit, $(a,b)$: open interval and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(ii), §32.4 and Ch.32

►

32.4.5

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Symbols:

$\mathrm{i}$: imaginary unit, $(a,b)$: open interval and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(ii), §32.4 and Ch.32

… ►

32.4.6

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $(a,b)$: open interval, $z$: real and $\alpha$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(iii), §32.4 and Ch.32

►

32.4.7

ⓘ

Symbols:

$\mathrm{i}$: imaginary unit, $(a,b)$: open interval and $z$: real

Permalink:

http://dlmf.nist.gov/32.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(iii), §32.4 and Ch.32

… ►

32.4.8

ⓘ

Symbols:

$(a,b)$: open interval, $z$: real and ${\theta }_{\mathrm{\infty }}$: constant

Permalink:

http://dlmf.nist.gov/32.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §32.4(iv), §32.4 and Ch.32

…

10: 26.6 Other Lattice Path Numbers

… ► $D(m,n)$ is the number of paths from $(0,0)$ to $(m,n)$ that are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. … ► $M(n)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$ and are composed of directed line segments of the form $(2,0)$, $(0,2)$, or $(1,1)$. … ► $N(n,k)$ is the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$, are composed of directed line segments of the form $(1,0)$ or $(0,1)$, and for which there are exactly $k$ occurrences at which a segment of the form $(0,1)$ is followed by a segment of the form $(1,0)$. … ► $r(n)$ is the number of paths from $(0,0)$ to $(n,n)$ that stay on or above the diagonal $y=x$ and are composed of directed line segments of the form $(1,0)$, $(0,1)$, or $(1,1)$. …