mathematical constants

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31: Bibliography K

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Separation of Variables for Riemannian Spaces of Constant Curvature. Longman Scientific & Technical, Harlow.

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External Links:

ISBN 0-582-98807-1, MathReview (H. Kilp), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

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Generalized functions. Mathematics in Science and Engineering, Vol. 171, Academic Press, Inc., Orlando, FL.

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Notes:

Theory and technique

External Links:

ISBN 0-12-396560-8, MathReview (Qing Yi Chen), ZentralBlatt

Referenced by:

§1.16(viii).

Encodings:

BibTeX

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Quantum Groups. Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York.

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External Links:

ISBN 0-387-94370-6, MathReview (Yu. N. Bespalov), ZentralBlatt

Referenced by:

§17.17.

Encodings:

BibTeX

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ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.

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Notes:

Double-precision Fortran, maximum accuracy 25S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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Jack, Hall-Littlewood and Macdonald Polynomials. Contemporary Mathematics, Vol. 417, American Mathematical Society, Providence, RI.

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Notes:

Proceedings of the workshop held in Edinburgh, September 23–26, 2003

External Links:

ISBN 0-8218-3683-8, MathReview Entry, ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

BibTeX

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32: 23.20 Mathematical Applications

§23.20 Mathematical Applications

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23.20.1 $$C:y^2=x^3+ax+b,$$

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Symbols:

$x$: real part of $z$ , $y$: imaginary part of $z$ , $C$: elliptic curve, $a$: constant and $b$: constant

Permalink:

http://dlmf.nist.gov/23.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.20(ii), §23.20 and Ch.23

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23.20.2 $$C:y^{2}z=x^3+ax{z}^2+b{z}^3,$$

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Symbols:

$x$: real part of $z$ , $y$: imaginary part of $z$ , $z$: complex, $C$: elliptic curve, $a$: constant and $b$: constant

Permalink:

http://dlmf.nist.gov/23.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.20(ii), §23.20 and Ch.23

… ►Here $a$ and $b$ are real or complex constants. … ►

33: 25.16 Mathematical Applications

§25.16 Mathematical Applications

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25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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34: 1.17 Integral and Series Representations of the Dirac Delta

… ►Other similar integral representations of the Dirac delta that appear in the physics and applied mathematics literature include the following: … ►

§1.17(iv) Mathematical Definitions

►The references given in §1.17(ii)–1.17(iii) are from the physics and applied mathematics literature. A comprehensive and detailed applied mathematics approach is that of Friedman (1990, Ch. 3 and 4 ). ►For mathematical derivations of the many of the results of §1.17(ii) and §1.17(iii) see Li and Wong (2008). …

35: Software Index

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	Open Source													With Book						Commercial
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5.24(ii) $\mathrm{\Gamma }\left(x\right)$, $x\in ℝ$	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	FDLIBM
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30.18(ii) Eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$														✓			✓		✓			✓
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Software published by the journal ACM Transactions on Mathematical Software (TOMS).

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A collection of mathematical software, papers, and databases produced by the numerical analysis research community.

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A cross index of mathematical software in use at NIST.

36: Bibliography O

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Tables of Lebedev, Mehler and Generalized Mehler Transforms. Mathematical Note Technical Report 246, Boeing Scientific Research Lab, Seattle.

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Referenced by:

§1.14(viii).

Encodings:

BibTeX

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Bessel Functions. Part III: Zeros and Associated Values. Royal Society Mathematical Tables, Volume 7, Cambridge University Press, Cambridge-New York.

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External Links:

ISBN 0-521-06153-9, MathReview (L. Fox), ZentralBlatt

Referenced by:

§10.21(vi), §10.21(viii), §10.58, §10.74(vi), 1st item, 2nd item.

Encodings:

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Tables for Bessel Functions of Moderate or Large Orders. National Physical Laboratory Mathematical Tables, Vol. 6. Department of Scientific and Industrial Research, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§10.20(i), §10.41(ii), 3rd item, 3rd item.

Encodings:

BibTeX

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The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.

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External Links:

MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§2.11(iii), §8.19(viii).

Encodings:

BibTeX

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Applications of Lie Groups to Differential Equations. 2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.

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External Links:

ISBN 0-387-94007-3; 0-387-95000-1, MathReview, ZentralBlatt

Referenced by:

§32.13(i).

Encodings:

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37: 1.4 Calculus of One Variable

… ►that is, for every arbitrarily small positive constant $ϵ$ there exists $\delta$ ($>0$) such that … ►If $f^{\prime }(x)\ge 0$ ($\le 0$) ($=0$) for all $x\in (a,b)$, then $f$ is nondecreasing (nonincreasing) (constant) on $(a,b)$. … ► … ► $c$ and $d$ constants. … ►see Rudin (1966), and often used in more abstract mathematical discussions. …