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1: Customize DLMF

…You can customize the appearance and functionality of the DLMF site using these selections. Note: Javascript and Cookies must be enabled to support customizations.

2: How to Cite

How to Cite

►When citing DLMF from a formal publication, we suggest a format similar to the following: …If you mention a specific equation (or chapter, section, …), you’ll help your readers find it by using our Permalinks & Reference numbers. … ►Citations from other electronic media (the web, email, …), should, of course, use the appropriate means to give the site URL (https://dlmf.nist.gov/), or specific Permalinks. … ►For convenience, the permalink can be found in the pop-up ‘Info box’ associated with each item in the site.

3: DLMF Project News

error generating summary

4: About MathML

… ►In rare cases, a browser lacks both MathML support and a robust enough javascript implementation capable of running MathJax; you may wish to visit the Customize DLMF page and choose the HTML+images document format. … ►Of course you are encouraged to use a modern, up-to-date browser. As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. … ►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site. DLMF uses the STIX web font to provide a consistent coverage. …

5: Mathematical Introduction

… ►In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …However, in many cases the coloring of the surface is chosen instead to indicate the quadrant of the plane to which the phase of the function belongs, thereby achieving a 4D effect. … ►In addition, the DLMF provides references to research papers in which software is developed, together with links to sites where the software can be obtained. … ►For all equations and other technical information this Handbook and the DLMF either provide references to the literature for proof or describe steps that can be followed to construct a proof. …

6: Foreword

… ► 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 provision of standard reference data of this type is a core function of NIST. … ►However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. …The DLMF may well serve as a model for the effective presentation of highly mathematical reference material on the Web. ►The production of these new resources has been a very complex undertaking some 10 years in the making. …

7: 18.40 Methods of Computation

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A numerical approach to the recursion coefficients and quadrature abscissas and weights

… ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain $w(x)$. … ►The question is then: how is this possible given only $F_N(z)$, rather than $F(z)$ itself? $F_N(z)$ often converges to smooth results for $z$ off the real axis for $\mathrm{\Im }z$ at a distance greater than the pole spacing of the $x_n$, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to $F_N(z)$ and evaluating these on the real axis in regions of higher pole density that those of the approximating function. … ►The quadrature points and weights can be put to a more direct and efficient use. … ►Interpolation of the midpoints of the jumps followed by differentiation with respect to $x$ yields a Stieltjes–Perron inversion to obtain $w^{\mathrm{RCP}}(x)$ to a precision of $\sim 4$ decimal digits for $N=120$. …

8: Guide to Searching the DLMF

… ►Every page contains a search box in the navigation bar. … ►

a textual word, a number, or a math symbol.

… ►If a query does not return any hits, the DLMF search system relaxes the query to match and retrieve what may be intended by the original query. … ►To recognize the math symbols and structures, and to accommodate equivalence between various notations and various forms of expression, the search system maps the math part of your queries into a different form. If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. …

9: 13.5 Continued Fractions

… ►If $a,b\in ℂ$ such that $a\ne -1,-2,-3,\mathrm{\dots }$, and $a-b\ne 0,1,2,\mathrm{\dots }$, then …This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). ►If $a,b\in ℂ$ such that $a\ne 0,-1,-2,\mathrm{\dots }$, and $b-a\ne 2,3,4,\mathrm{\dots }$, then …This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

10: About Color Map

… ►To provide an easily interpreted encoding of surface heights, a rainbow-like mapping of height to color is used. … ►By painting the surfaces with a color that encodes the phase, $\mathrm{ph}f$, both the magnitude and phase of complex valued functions can be displayed. … ►As a mnemonic, the colors are sorted alphabetically. … ►In doing this, however, we would like to place the mathematically significant phase values, specifically the multiples of $\pi /2$ correponding to the real and imaginary axes, at more immediately recognizable colors. … ►We therefore use a piecewise linear mapping as illustrated below, that takes phase $0$ to red, $\pi /2$ to yellow, $\pi$ to cyan and $3\pi /2$ to blue. …