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11: Notices

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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.

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12: Bibliography U

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Unpublished Mathematical Tables (1944) Mathematics of Computation Unpublished Mathematical Tables Collection.

ⓘ

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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13: 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. …

14: Bibliography M

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I. G. Macdonald (1990) Hypergeometric Functions.

ⓘ

Notes:: Unpublished Lecture Notes, University of London.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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Maple (commercial interactive system) Maplesoft.

ⓘ

Notes:: General purpose mathematical computing environment. Includes more than 100 special functions. Includes symbolic and arbitrary precision numerical computation.
Referenced by:: 1st item, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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L. C. Maximon (1991) 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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J. Miller and V. S. Adamchik (1998) Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.

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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: (25.11.21), (25.11.22), (25.11.23), §25.11, §25.11(vi), §25.18(i), (25.6.15).
Encodings:: BibTeX

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G. W. Morgenthaler and H. Reismann (1963) Zeros of first derivatives of Bessel functions of the first kind,

J_{n}^{\prime}\,(x)

21\leq n\leq 51

0\leq x\leq 100

. J. Res. Nat. Bur. Standards Sect. B 67B (3), pp. 181–183.

ⓘ

External Links:: ISSN 0160-1741, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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15: Bibliography L

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D. W. Lozier and F. W. J. Olver (1994) Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.

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Notes:: An updated version exists online.
External Links:: FullText, MathReview (John P. Coleman), ZentralBlatt
Referenced by:: Ch.3.
Encodings:: BibTeX

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Y. L. Luke and J. Wimp (1963) Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.

ⓘ

External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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16: 18.39 Applications in the Physical Sciences

… ►where

x

is a spatial coordinate,

m

the mass of the particle with potential energy

V(x)

\hbar=h/(2\pi)

is the reduced Planck’s constant, and

(a,b)

a finite or infinite interval. ►Here the term

-\frac{\hbar^{2}}{2m}\tfrac{{\partial}^{2}}{{\partial x}^{2}}

represents the quantum kinetic energy of a single particle of mass

m

, and

V(x)

its potential energy. … ►and

\hbar=k=m=1

, has eigenfunctions … ►The eigenfunctions of

\mathrm{L}^{2}

are the spherical harmonics

Y_{{l},{m_{l}}}\left(\theta,\phi\right)

with eigenvalues

\hbar^{2}l(l+1)

, each with degeneracy

2l+1

m_{l}=-l,-l+1,\dots,l

. … ►,

\hbar=m_{e}={e}^{2}=4\pi\epsilon_{0}=1

, Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of

a . u .

to SI units is spelled out. …

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

\left\langle u,v\right\rangle\in\mathbb{C}

is defined for all

u,v\in V

with the properties: (i)

\left\langle u,v\right\rangle

is complex linear in

u

; (ii)

\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}

; (iii)

\left\langle v,v\right\rangle\geq 0

; (iv) if

\left\langle v,v\right\rangle=0

then

v=0

. …Two elements

u

and

v

V

are orthogonal if

\left\langle u,v\right\rangle=0

. … ►Functions

f,g\in L^{2}\left(X,\,\mathrm{d}\alpha\right)

for which

\left\langle f-g,f-g\right\rangle=0

are identified with each other. … ►,

\left\langle u_{\lambda},u_{\lambda^{\prime}}\right\rangle=0

, for

\lambda\neq\lambda^{\prime}

. … ►The adjoint

{T}^{*}

T

does satisfy

\left\langle Tf,g\right\rangle=\left\langle f,{T}^{*}g\right\rangle

where

\left\langle f,g\right\rangle=\int_{a}^{b}f(x)g(x)\,\mathrm{d}x

. …

18: 1.2 Elementary Algebra

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1.2.40

\left\langle\mathbf{u},\mathbf{v}\right\rangle=\sum_{i=1}^{n}u_{i}\overline{v_% {i}}={\mathbf{v}}^{{\rm H}}\mathbf{u}.

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Defines:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors
Symbols:: ${\NVar{\mathbf{A}}}^{{\rm H}}$ : Hermitian conjugate of matrix, $\overline{\NVar{z}}$ : complex conjugate, $n$ : nonnegative integer, $\mathbf{v}$ : column vector and $\mathbf{u}$ : 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

\left\langle\mathbf{u},\mathbf{v}\right\rangle=\overline{\left\langle\mathbf{v% },\mathbf{u}\right\rangle},

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : 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

\left\langle\alpha\mathbf{u},\beta\mathbf{v}\right\rangle=\alpha\overline{% \beta}\left\langle\mathbf{u},\mathbf{v}\right\rangle,

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : 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

\left\langle\mathbf{v},\mathbf{v}\right\rangle=0,

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Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors and $\mathbf{v}$ : 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

… ►

1.2.44

\left\langle\mathbf{u},\mathbf{v}\right\rangle=0.

ⓘ

Symbols:: $\left\langle\NVar{\mathbf{u}},\NVar{\mathbf{v}}\right\rangle$ : inner product over vectors, $\mathbf{v}$ : column vector and $\mathbf{u}$ : 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: 1.1 Special Notation

… ► ►►►

$x, y$	real variables.
…
$\left\langle f,g\right\rangle$	inner, or scalar, product for real or complex vectors or functions.
…

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