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11: Errata

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Subsection 25.10(ii)

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).

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Equation (3.3.34)

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

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Chapter 35 Functions of Matrix Argument

The generalized hypergeometric function of matrix argument ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ , was linked inadvertently as its single variable counterpart ${{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)$ . Furthermore, the Jacobi function of matrix argument $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and the Laguerre function of matrix argument $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ , were also linked inadvertently (and incorrectly) in terms of the single variable counterparts given by $P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)$ , and $L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)$ . In order to resolve these inconsistencies, these functions now link correctly to their respective definitions.

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Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . 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 $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

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

Reported 2014-02-28 by Svante Janson.

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References

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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12: Foreword

… ►In 1964 the National Institute of Standards and Technology¹¹ 1 Then known as the National Bureau of Standards. published the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, edited by Milton Abramowitz and Irene A. … ►The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium. …

13: 3.12 Mathematical Constants

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3.12.2

\pi=4\int_{0}^{1}\frac{\,\mathrm{d}t}{1+t^{2}}.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral
Permalink:: http://dlmf.nist.gov/3.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §3.12 and Ch.3

… ►For access to online high-precision numerical values of mathematical constants see Sloane (2003). …

14: 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. …

15: About the Project

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Refer to caption — Figure 1: The Editors and 9 of the 10 Associate Editors of the DLMF Project (photo taken at 3rd Editors Meeting, April, 2001). …

… ►These products resulted from the leadership of the Editors and Associate Editors pictured in Figure 1; the contributions of 29 authors, 10 validators, and 5 principal developers; and assistance from a large group of contributing developers, consultants, assistants and interns. … ►They were selected as recognized leaders in the research communities interested in the mathematics and applications of special functions and orthogonal polynomials; in the presentation of mathematics reference information online and in handbooks; and in the presentation of mathematics on the web. …

16: Bibliography U

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H. Umemura and H. Watanabe (1998) Solutions of the third Painlevé equation. I. Nagoya Math. J. 151, pp. 1–24.

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External Links:: ISSN 0027-7630, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.10(iii).
Encodings:: BibTeX

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Unpublished Mathematical Tables (1944) Mathematics of Computation Unpublished Mathematical Tables Collection.

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Notes:: Archives of American Mathematics, Center for American History, The University of Texas at Austin. Covers 1944–1994. An online guide is arranged chronologically by date of issuance within Mathematics of Computation
External Links:: Guide
Referenced by:: §27.21.
Encodings:: BibTeX

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J. Urbanowicz (1988) On the equation

f(1)1^{k}+f(2)2^{k}+\cdots+f(x)x^{k}+{R}(x)={B}y^{2}

. Acta Arith. 51 (4), pp. 349–368.

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External Links:: ISSN 0065-1036, Pdf, MathReview (István Gaál), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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17: Frank W. J. Olver

… ►g. … ►Most notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). … ►

1 Algebraic and Analytic Methods

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18: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Two elements

u

and

v

V

are orthogonal if

\left\langle u,v\right\rangle=0

. … ►Functions

f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)

for which

\left\langle f-g,f-g\right\rangle=0

are identified with each other. … ►The adjoint

{T}^{*}

T

does satisfy

\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle

where

\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x

. … ►where

\widehat{f}(\lambda_{n})

\frac{1}{\sqrt{\pi}}\int_{0}^{\pi}f(y){\mathrm{e}}^{-2\mathrm{i}ny}\,\mathrm{d% }y=\left\langle f,\phi_{\mathrm{exp}}(n)\right\rangle

, being that of (1.8.3) and (1.8.4). … ►This dilatation transformation, which does require analyticity of

q(x)

in (1.18.28), or an analytic approximation thereto, leaves the poles, corresponding to the discrete spectrum, invariant, as they are, as is the branch point, actual singularities of

\left\langle\left(z-T\right)^{-1}f,f\right\rangle

. …

19: DLMF Project News

error generating summary

20: 1.2 Elementary Algebra

… ►where

\ell

n

n-1

according as

n

is even or odd. … ►where

\ell

= last term of the series =

a+(n-1)d

. … ►The geometric mean

G

and harmonic mean

H

n

positive numbers

a_{1},a_{2},\dots,a_{n}

are given by … ►A column vector of length

n

is an

n\times 1

matrix … ►the

l^{1}

norm …