DLMF: Untitled Document

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Fettis et al. (1973) gives the first 100 zeros of $\operatorname{erf}z$ and $w\left(z\right)$ (the table on page 406 of this reference is for $w\left(z\right)$ , not for $\operatorname{erfc}z$ ), 11S.

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PARI-GP (free interactive system and C library)

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Notes:: Computer algebra system designed for computations in number theory. Many transcendental functions are also included. PARI is also available as a C library to allow for faster computations. Originally developed by Henri Cohen and co-workers at the Université Bordeaux I.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

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Prime Pages (website)

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Notes:: Information on primes, primality testing, and factorization including links to programs and lists of primes.
External Links:: Home Page
Referenced by:: 7th item.
Encodings:: BibTeX

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R. L. Hall, N. Saad, and K. D. Sen (2010) Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.

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External Links:: ISSN 0022-2488, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §31.17(ii), §31.17(ii).
Encodings:: BibTeX

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H. J. Haubold, A. M. Mathai, and R. K. Saxena (2011) Mittag-Leffler functions and their applications. J. Appl. Math. 2011, pp. Art. ID 298628, 51 pages.

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External Links:: ISSN 1110-757X, Document, FullText, MathReview (Rudolf Gorenflo), ZentralBlatt
Referenced by:: §10.46, §10.46.
Encodings:: BibTeX

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R. S. Heller (1976) 25D Table of the First One Hundred Values of

j_{0,s}

,

J_{1}(j_{0,s})

,

j_{1,s}

,

J_{0}(j_{1,s})=J_{0}(j^{\prime}_{0,s+1})

,

j^{\prime}_{1,s}

,

J_{1}(j^{\prime}_{1,s})

. Technical report Department of Physics, Worcester Polytechnic Institute, Worcester, MA.

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Notes:: Ms. of six pages deposited in the UMT file
Referenced by:: 8th item.
Encodings:: BibTeX

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Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:: Includes hypergeometric and $q$ -hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).
Encodings:: BibTeX

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13.6.11_1

M\left(\nu+\tfrac{1}{2},2\nu+1+n,2z\right)=\Gamma\left(\nu\right){\mathrm{e}}^% {z}\left(\ifrac{z}{2}\right)^{-\nu}\sum_{k=0}^{n}\frac{{\left(-n\right)_{k}}{% \left(2\nu\right)_{k}}(\nu+k)}{{\left(2\nu+1+n\right)_{k}}k!}I_{\nu+k}\left(z% \right),

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $n$ : nonnegative integer and $z$ : complex variable
Source:: Prudnikov et al. (1990, page 579, (7.11.1.6))
Referenced by:: §13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/13.6.E11_1
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
See also:: Annotations for §13.6(iii), §13.6 and Ch.13

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13.6.11_2

M\left(\nu+\tfrac{1}{2},2\nu+1-n,2z\right)=\Gamma\left(\nu-n\right){\mathrm{e}% }^{z}\left(\ifrac{z}{2}\right)^{n-\nu}\sum_{k=0}^{n}(-1)^{k}\frac{{\left(-n% \right)_{k}}{\left(2\nu-2n\right)_{k}}(\nu-n+k)}{{\left(2\nu+1-n\right)_{k}}k!% }I_{\nu+k-n}\left(z\right).

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $n$ : nonnegative integer and $z$ : complex variable
Source:: Prudnikov et al. (1990, page 579, (7.11.1.7))
Referenced by:: §13.11, §13.6(iii), §13.6(iii), Erratum (V1.1.0) for Additions
Permalink:: http://dlmf.nist.gov/13.6.E11_2
Encodings:: pMML, png, TeX
Addition (effective with 1.1.0):: This equation was added.
See also:: Annotations for §13.6(iii), §13.6 and Ch.13

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REDUCE (free interactive system)

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Notes:: Symbolic and extended-precision general purpose computer algebra system.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

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H. P. Robinson (1972) Roots of

\tan x=x

.

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Notes:: 10 typewritten pages deposited in the UMT file at Lawrence Berkeley Laboratory, University of California, Berkeley, CA.
Referenced by:: §4.46.
Encodings:: BibTeX

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SAGE (free interactive system)

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Notes:: SAGE is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open-source packages into a common Python-based interface.
External Links:: Home Page
Referenced by:: § ‣ 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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J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:: For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.
External Links:: Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt
Referenced by:: 3rd item.
Encodings:: BibTeX

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Stembridge (website) John Stembridge’s Home Page

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Notes:: Includes Maple software for zonal, Jack, and Macdonald polynomials; see the SF Package.
External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

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P. Koev and A. Edelman (2006) The efficient evaluation of the hypergeometric function of a matrix argument. Math. Comp. 75 (254), pp. 833–846.

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Notes:: For Matlab software for this function, see Plamen Koev’s home page.
External Links:: Home Page, ISSN 0025-5718, Document, MathReview (Luis Verde-Star), ZentralBlatt
Referenced by:: §35.10, 2nd item.
Encodings:: BibTeX

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Koornwinder (website) Tom Koornwinder’s Personal Collection of Maple Procedures

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External Links:: Home Page
Referenced by:: § ‣ Software Cross Index.
Encodings:: BibTeX

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… ►The possibility of generalization to

\alpha=-k

, for

k\in\mathbb{N}

, is implicit in the identity Szegő (1975, page 102), …

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ECMNET Project (website)

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Notes:: Links to software for elliptic curve methods of factorization and primality testing.
External Links:: Home Page
Referenced by:: 4th item.
Encodings:: BibTeX

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11—20 of 31 matching pages

11: 7.23 Tables

12: Bibliography P

13: Bibliography H

14: Bibliography Z

15: 13.6 Relations to Other Functions

16: Bibliography R

17: Bibliography S

18: Bibliography K

19: 18.36 Miscellaneous Polynomials

20: Bibliography E

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