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1: 8.24 Physical Applications

… ►The function $\gamma (a,x)$ appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)). … ►The function $I_x(a,b)$ appears in: Monte Carlo sampling in statistical mechanics (Kofke (2004)); analysis of packings of soft or granular objects (Prellberg and Owczarek (1995)); growth formulas in cosmology (Hamilton (2001)). … ►The function $E_p\left(x\right)$, with $p>0$, appears in theories of transport and radiative equilibrium (Hopf (1934), Kourganoff (1952), Altaç (1996)). ►With more general values of $p$, $E_p\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).

2: Bibliography D

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Asymptotic Methods in Analysis. 2nd edition, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§2.2, Ch.2, §26.7(iv), §26.7(iv), §4.13.

Encodings:

BibTeX

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Atomic motions in a rigid sphere gas as a problem in neutron transport. Nucl. Sci. Eng. 24 (2), pp. 142–152.

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Referenced by:

§18.39(iii).

Encodings:

BibTeX

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Bernoulli Numbers. Bibliography (1713–1990). Queen’s Papers in Pure and Applied Mathematics, Vol. 87, Queen’s University, Kingston, ON.

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Notes:

A frequently updated version, with links to reviews, exists online

External Links:

Online version, MathReview, ZentralBlatt

Referenced by:

Ch.24.

Encodings:

BibTeX

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A simplified algorithm for the second-order sound fields. J. Acoust. Soc. Amer. 108 (6), pp. 2759–2764.

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External Links:

Document

Referenced by:

§8.24(iii).

Encodings:

BibTeX

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Ramanujan’s master theorem for symmetric cones. Pacific J. Math. 175 (2), pp. 447–490.

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External Links:

ISSN 0030-8730, MathReview (Jacques Faraut), ZentralBlatt

Referenced by:

§35.7(ii), §35.8(iv), §35.8(v).

Encodings:

BibTeX

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3: Possible Errors in DLMF

Possible Errors in DLMF

►We have strived for the utmost in correctness, clarity and accuracy, having validated each chapter with both internal and external experts. ►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. …

4: How to Cite

… ►For example, the sentence in the LaTeX source: …(assuming \usepackage{hyperref} in the preamble, and NIST:DLMF being the key for DLMF) … ►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. … ►For convenience, the permalink can be found in the pop-up ‘Info box’ associated with each item in the site.

5: 3.5 Quadrature

… ►Similar results hold for the trapezoidal rule in the form …In Stenger (1993, Chapter 3) the rule (3.5.5) is considered in the framework of Sinc approximations (§3.3(vi)). … ►These also immediately yield the recurrence coefficients in (3.5.30_5). …In particular, by continuity in $\alpha$ and $\beta$, … ►All digits shown in the approximation in the final row are correct. …

6: 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. … ►WebGL is supported in the current versions of most common web browsers. ►While WebGL reportedly works with Internet Explorer 11 in Windows 8. … ►In fact, we have not found a VRML/X3D browser or plugin that works consistently with our VRML/X3D visualization files in Linux. Linux users must view the visualizations in a WebGL-enabled browser. …

7: 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. … ►Much has changed in the years since A&S was published. Certainly, advances in applied mathematics have continued unabated. …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. … ►The production of these new resources has been a very complex undertaking some 10 years in the making. …

8: 3.12 Mathematical Constants

… ►can be defined analytically in numerous ways, for example, …Other constants that appear in the DLMF include the base $\mathrm{e}$ of natural logarithms … ►For access to online high-precision numerical values of mathematical constants see Sloane (2003). …

9: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►Let $p$ and $q$ be nonnegative integers; $a_1,\mathrm{\dots },a_p\in ℂ$; $b_1,\mathrm{\dots },b_q\in ℂ$; $-b_j+\frac{1}{2}(k+1)\notin \mathbb{N}$, $1\le j\le q$, $1\le k\le m$. The generalized hypergeometric function ${}_p{}^{}F_q^{}$ with matrix argument $𝐓\in 𝓢$, numerator parameters $a_1,\mathrm{\dots },a_p$, and denominator parameters $b_1,\mathrm{\dots },b_q$ is … ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions ${}_p{}^{}F_q^{}$ and ${}_{p+1}{}^{}F_p^{}$ of matrix argument. A similar result for the ${}_0{}^{}F_1^{}$ function of matrix argument is given in Faraut and Korányi (1994, p. 346). These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$. …

10: Notices

… ►Certain products, commercial and otherwise, are mentioned in the DLMF. … ►If you feel you have found an error in DLMF, please see Possible Errors in DLMF. … ►Using special functions in applications often requires computing them. …We do this in two ways. ►

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.

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