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11: Notices

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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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12: Philip J. Davis

… ►In 1957, Davis took over as Chief, Numerical Analysis Section when John Todd and his wife Olga Taussky-Todd, feeling a strong pull toward teaching and research, left to pursue full-time positions at the California Institute of Technology. … ►NBS mathematician Irene Stegun took over management of the A&S project which was already well on its way, and led the work to publication in 1964. … ►Davis’s comments about our uninspired graphs sparked the research and design of techniques for creating interactive 3D visualizations of function surfaces, which grew in sophistication as our knowledge and the technology for developing 3D graphics on the web advanced over the years. Today the DLMF contains close to 600 2D and 3D graphs and more than 200 interactive 3D visualizations. …

13: Bibliography U

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Mathematics of Computation Unpublished Mathematical Tables Collection.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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14: Frank W. J. Olver

… ►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). …

15: About the Project

… ►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 O

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Computing $\pi (x)$: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:

Online fulltext contains postpublication annotations.

External Links:

FullText

Referenced by:

§27.18.

Encodings:

BibTeX

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Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

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Notes:

Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

Encodings:

BibTeX

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17: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space $V$ is called an inner product space if an inner product $⟨u,v⟩\in ℂ$ is defined for all $u,v\in V$ with the properties: (i) $⟨u,v⟩$ is complex linear in $u$; (ii) $⟨u,v⟩=\overline{⟨v,u⟩}$; (iii) $⟨v,v⟩\ge 0$; (iv) if $⟨v,v⟩=0$ then $v=0$. …Two elements $u$ and $v$ in $V$ are orthogonal if $⟨u,v⟩=0$. … ►Functions $f,g\in L^2(X,d\alpha )$ for which $⟨f-g,f-g⟩=0$ are identified with each other. … ►, $⟨u_{\lambda },u_{{\lambda }^{\prime }}⟩=0$, for $\lambda \ne {\lambda }^{\prime }$. … ►The adjoint $T^{\ast }$ of $T$ does satisfy $⟨Tf,g⟩=⟨f,T^{\ast }g⟩$ where $⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$. …

18: 1.2 Elementary Algebra

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1.2.40 $$⟨𝐮,𝐯⟩=\sum _{i=1}^n{u}_i\overline{v_i}={𝐯}^{\mathrm{H}}𝐮.$$

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Defines:

$⟨𝐮,𝐯⟩$: inner product over vectors

Symbols:

${𝐀}^{\mathrm{H}}$: Hermitian conjugate of matrix, $\overline{z}$: complex conjugate, $n$: nonnegative integer, $𝐯$: column vector and $𝐮$: column vector

Referenced by:

§1.18(i), §1.2(v)

Permalink:

http://dlmf.nist.gov/1.2.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.41 $$⟨𝐮,𝐯⟩=\overline{⟨𝐯,𝐮⟩},$$

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Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.42 $$⟨\alpha 𝐮,\beta 𝐯⟩=\alpha \overline{\beta }⟨𝐮,𝐯⟩,$$

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Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Referenced by:

§1.2(v)

Permalink:

http://dlmf.nist.gov/1.2.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.43 $$⟨𝐯,𝐯⟩=0,$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors and $𝐯$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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1.2.44 $$⟨𝐮,𝐯⟩=0.$$

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Symbols:

$⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Permalink:

http://dlmf.nist.gov/1.2.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

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19: Karl Dilcher

… ►Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. …

20: Bibliography M

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Hypergeometric Functions.

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Notes:

Unpublished Lecture Notes, University of London. An online version exists.

External Links:

FullText

Referenced by:

§35.7(ii), §35.8(iii).

Encodings:

BibTeX

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On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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External Links:

ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

Encodings:

BibTeX

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