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1: 25.10 Zeros

… ►Calculations relating to the zeros on the critical line make use of the real-valued function …is chosen to make

Z(t)

real, and

\operatorname{ph}\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)

assumes its principal value. … ►Sign changes of

Z(t)

are determined by multiplying (25.9.3) by

\exp\left(i\vartheta(t)\right)

to obtain the Riemann–Siegel formula: …where

R(t)=O\left(t^{-1/4}\right)

t\to\infty

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

… ►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. There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

3: How to Cite

How to Cite

►When citing DLMF from a formal publication, we suggest a format similar to the following: … ►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. Note the ‘E’, ‘F’ and ‘T’ used to disambiguate equations, figures and tables. …

4: Notices

… ►Pursuant to Title 17 USC 105, the National Institute of Standards and Technology (NIST), United States Department of Commerce, is authorized to receive and hold copyrights transferred to it by assignment or otherwise. … ►Questions regarding this copyright policy should be directed to dlmf-feedback. … ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. …Thus, we seek to provide DLMF users with links to sources of such software. … ►

Index of Selected Software Within the DLMF Chapters

Within each of the DLMF chapters themselves we will provide a list of research software for the functions discussed in that chapter. The purpose of these listings is to provide references to the research literature on the engineering of software for special functions. To qualify for listing, the development of the software must have been the subject of a research paper published in the peer-reviewed literature. If such software is available online for free download we will provide a link to the software.

In general, we will not index other software within DLMF chapters unless the software is unique in some way, such as being the only known software for computing a particular function.

…

5: Viewing DLMF Interactive 3D Graphics

… ►Below we provide some notes and links to online material which might be helpful in viewing our visualizations, but please see our Disclaimer. We have strived to provide the most current information available, but if any details appear to be out of date or confusing, please notify us. … ►For that reason we will continue to offer VRML/X3D options as indicated under Additional Options. … ►If you prefer to use a web browser that is not WebGL-enabled, we currently also provide VRML and X3D versions of the DLMF visualizations that may be viewed with a special VRML/X3D browser or plugin. …Both are expected to be superseded by WebGL. …

6: Foreword

… ►That 1046-page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their day-to-day work, so much so that it became the most widely distributed and most highly cited NIST publication in the first 100 years of the institution’s existence. ²² 2 D. R. Lide (ed.), A Century of Excellence in Measurement, Standards, and Technology, CRC Press, 2001. The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. … ►The document you are now holding, or the Web page you are now reading, represents an effort to extend the legacy of A&S well into the 21st century. …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. … ►Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. …

7: 3.12 Mathematical Constants

… ►

3.12.1

\pi=3.14159\;26535\;89793\;23846\;\ldots

ⓘ

Defines:: $\pi$ : the ratio of the circumference of a circle to its diameter
Notes:: For more digits see OEIS Sequence A000796; see also Sloane (2003).
Permalink:: http://dlmf.nist.gov/3.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §3.12 and Ch.3

… ►

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). …

8: Frank W. J. Olver

… ►In April 2011, NIST co-organized a conference on “Special Functions in the 21st Century: Theory & Application” which was dedicated to Olver. … ►Olver continued to maintain a connection to NIST after moving to the university. 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). Fitting capstones to Olver’s long career, these works are on track to extend the legacy of the classic Abramowitz and Stegun handbook well into the 21st century. According to DLMF Project Lead Daniel Lozier, “Olver’s encyclopedic knowledge of the field, his clear vision for mathematical exposition, his keen sense of the needs of practitioners, and his unfailing attention to detail were key to the success of that project. …

9: About the Project

… ►Subsequently, the list of Editors has been expanded to include Bruce R. … ►The former title of Associate Editor has been changed to Senior Associate Editor. …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. Their responsibility is to advise the Editors on future directions for the DLMF. … ►They will be called upon to help deal with reports of suspected errors and suggestions for additions or other modifications to the chapters. …

10: DLMF Project News

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