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1: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(xi) Further Generalizations

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2: Need Help?

… ►In the Digital Library of Mathematical Functions, we have tried to provide the most accurate, carefully selected information about Special Functions possible. We have also tried to use the best technologies available in order to make this information useful and accessible. … ►

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• See also the Index or Notations sections.

• Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

3: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ►If none of the $a_j$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to $z$. The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$; compare §4.2(i). … ►can be used to interchange $p$ and $q$. … ►

§16.2(v) Behavior with Respect to Parameters

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4: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

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§8.21(i) Definitions: General Values

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§8.21(v) Special Values

… ►When $z\to \mathrm{\infty }$ with $|\mathrm{ph}z|\le \pi -\delta$ (), … ►

5: 1.16 Distributions

… ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►We denote a regular distribution by ${\mathrm{\Lambda }}_f$, or simply $f$, where $f$ is the function giving rise to the distribution. …More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: … ►The second to last equality follows from the Fourier integral formula (1.17.8). … ►Friedman (1990) gives an overview of generalized functions and their relation to distributions. …

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. 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. … ►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 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. … ►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: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(i) Definition

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§35.8(ii) Relations to Other Functions

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§35.8(iv) General Properties

… ►These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$. …

8: About MathML

… ►, built-in to the browser) support for MathML is growing, (see Browsers supporting MathML). However, the MathJax javascript library can also used to render the MathML, albeit more slowly. By default, DLMF will use Native support when available; You may choose how MathML is processed (Native or MathJax) at Customize DLMF. … ►Of course you are encouraged to use a modern, up-to-date browser. … ►Recent enhancements to the WebKit engine now provide support for MathML Core. …

9: How to Cite

How to Cite

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

10: DLMF Project News

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