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1: 26.15 Permutations: Matrix Notation

… ►where the sum is over and $n\ge h>\mathrm{\ell }\ge 1$. … ►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. ►

26.15.5 $$R(x,B)=xR(x,B\setminus [j,k])+R(x,B\setminus (j,k)).$$

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Symbols:

$[a,b]$: closed interval, $(a,b)$: open interval, $\setminus$: set subtraction, $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $B$: subset and $R(x,B)$: rook polynomial

Permalink:

http://dlmf.nist.gov/26.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15 and Ch.26

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2: 21.7 Riemann Surfaces

… ►where $P(\lambda ,\mu )$ is a polynomial in $\lambda$ and $\mu$ that does not factor over ${ℂ}^2$. … ►where $Q(\lambda )$ is a polynomial in $\lambda$ of odd degree $2g+1$ $(\ge 5)$. … ►

21.7.12 $$T_1\ominus T_2=(T_1\cup T_2)\setminus (T_1\cap T_2).$$

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Symbols:

$\cap$: intersection, $\setminus$: set subtraction, $\cup$: union and $T$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

►Also, $T^c=B\setminus T$, ${𝜼}^1(T)=({\eta }_1(T),{\eta }_2(T),\mathrm{\dots },{\eta }_g(T))$, and ${𝜼}^2(T)=({\eta }_{g+1}(T),{\eta }_{g+2}(T),\mathrm{\dots },{\eta }_{2g}(T))$. …

3: 36.5 Stokes Sets

… ►

36.5.2 $$y^3=\frac{27}{4}\left(\sqrt{27}-5\right)x^2=1.32403x^2.$$

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Symbols:

$y$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►

$x=B_{\pm }{|y|}^{4/3},$

►

$B_{\pm }={10}^{-1/3}\left(2x_{\pm }^{4/3}-\frac{1}{2}{x}_{\pm }^{-2/3}\right),$

… ►

36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►For , there are two solutions $u$, provided that $|Y|>{(\frac{2}{5})}^{1/2}$. …

4: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

5: Bibliography B

… ►

Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:

ISSN 0893-9659, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

On numerical evaluation of integrals involving Bessel functions. Apl. Mat. 31 (5), pp. 396–410.

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External Links:

ISSN 0373-6725, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

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External Links:

ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

Encodings:

BibTeX

… ►

Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.

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Notes:

Single-precision Fortran, maximum accuracy 16S

External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

Encodings:

BibTeX

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6: Errata

… ►

In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

… ►

In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

… ►

8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right)$$

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris.

… ►

Originally a minus sign was missing in the entries for $\mathrm{cd}u$ and $\mathrm{dc}u$ in the second column (headed $z+K+i{K}^{\prime }$). The correct entries are $-k^{-1}\mathrm{ns}z$ and $-k\mathrm{sn}z$. Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}{K}^{\prime }$	$z+\mathrm{i}{K}^{\prime }$	$z+2K$	$z+2K+2\mathrm{i}{K}^{\prime }$	$z+2\mathrm{i}{K}^{\prime }$
$\mathrm{⋮}$
$\mathrm{cd}u$	$-\mathrm{sn}z$	$-k^{-1}\mathrm{ns}z$	$k^{-1}\mathrm{dc}z$	$-\mathrm{cd}z$	$-\mathrm{cd}z$	$\mathrm{cd}z$
$\mathrm{⋮}$
$\mathrm{dc}u$	$-\mathrm{ns}z$	$-k\mathrm{sn}z$	$k\mathrm{cd}z$	$-\mathrm{dc}z$	$-\mathrm{dc}z$	$\mathrm{dc}z$
$\mathrm{⋮}$

Reported 2014-02-28 by Svante Janson.

… ►

Bibliographic citations were added in §§3.5(iv), 4.44, 8.22(ii), 22.4(i), and minor clarifications were made in §§19.12, 20.7(vii), 22.9(i). In addition, several minor improvements were made affecting only ancilliary documents and links in the online version.

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7: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the icon) for links to defining formula. …Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

8: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►

${\alpha }_5=\frac{1}{4320},$

… ►

$c_5(0)=-\frac{\mathrm{27\hspace{0.33em}45493}}{\mathrm{81517\hspace{0.33em}36320}}.$

… ►

8.12.18 $$\begin{array}{c}Q(a,z)\\ P(a,z)\end{array}\}\sim \frac{z^{a-\frac{1}{2}}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(a\right)}\left(d(\pm \chi )\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\chi )}{z^{k/2}}\mp \sum _{k=1}^{\mathrm{\infty }}\frac{B_k(\chi )}{z^{k/2}}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\chi$ , $d(\pm \chi )$ , $A_k(\chi )$ and $B_k(\chi )$

Referenced by:

§8.12, Erratum (V1.0.16) for Equation (8.12.18)

Permalink:

http://dlmf.nist.gov/8.12.E18

Encodings:

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Errata (effective with 1.0.16):

The original $\pm$ in front of the second summation was replaced by $\mp$ to correct an error in Paris (2002b); for details see https://arxiv.org/abs/1611.00548.

Reported 2017-01-28 by Richard Paris

See also:

Annotations for §8.12 and Ch.8

…

9: How to Cite

… ►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. … ► ►►►

Item Type	Ref. Number	Permalink
Chapters	5	https://dlmf.nist.gov/5
…

…

10: 19.16 Definitions

… ►

19.16.7 $$\frac{3}{2}{\int }_0^{\mathrm{\infty }}\frac{dt}{{\prod }_{j=1}^5\sqrt{t+x_j}}.$$

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Symbols:

$dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/19.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.16(i), §19.16 and Ch.19

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