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1: 15.12 Asymptotic Approximations

… ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

2: Publications

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Design of a Digital Mathematical Library for Science, Technology and Education, Proceedings of the IEEE Forum on Research and Technology Advances in Digital Libraries (IEEE ADL ’99, Baltimore, Maryland, May 19, 1999).

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Web-Based 3D Visualization in a Digital Library of Mathematical Functions, Proceedings of the Web3D Symposium, Bangor, UK, March 29–April 1, 2005.

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From B-Spline Mesh Generation to Effective Visualizations for the NIST Digital Library of Mathematical Functions, in Curve and Surface Design, Proceedings of the Sixth International Conference on Curves and Surfaces, Avignon, France June 29–July 5, 2006, pp. 235–243.

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Creating Webs of Math Using LaTeX, Proceedings 6th International Congress on Industrial and Applied Mathematics, Zürich, Switzerland, July 17, 2007.

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A Special Functions Handbook for the Digital Age, Notices of the American Mathematical Society 58, 7 (2011), pp. 905–911.

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3: Bibliography C

Bibliography C

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Inequalities for a symmetric elliptic integral. Proc. Amer. Math. Soc. 25 (3), pp. 698–703.

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External Links:

FullText, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.24(ii), §19.9(ii).

Encodings:

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Toward symbolic integration of elliptic integrals. J. Symbolic Comput. 28 (6), pp. 739–753.

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Notes:

Code written by J. FitzSimons to generate reduction tables for elliptic integrals symbolically in the Derive computer algebra system is available.

External Links:

ISSN 0747-7171, Document, Code, MathReview (Axel Riese), ZentralBlatt

Referenced by:

§19.29(i), §19.29(ii), §19.29(ii), §19.29(ii), §19.29(ii).

Encodings:

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Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).

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External Links:

MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§19.31.

Encodings:

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An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.

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Notes:

Republished by Dover Publications, Inc,, New York, 1961. Table erratum: Math. Comp. v. 29 (1975), no. 130, p. 670.

External Links:

ZentralBlatt

Referenced by:

§19.12, Ch.19, A. Cayley (1961).

Encodings:

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4: Staff

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Ronald F. Boisvert, Editor at Large, NIST

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Hans Volkmer, University of Wisconsin, Milwaukee, Chaps. 29, 30

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Gerhard Wolf, University of Duisberg-Essen, Chap. 28

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Simon Ruijsenaars, University of Leeds, for Chaps. 5, 28

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Hans Volkmer, University of Wisconsin–Milwaukee, for Chaps. 29, 30

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5: Bibliography B

Bibliography B

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Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 5, 109–117

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.77(iv).

Encodings:

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Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7, pp. 1620–1636.

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External Links:

Document

Referenced by:

§22.19(ii).

Encodings:

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Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.5(iv).

Encodings:

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Two cold atoms in a harmonic trap. Found. Phys. 28 (4), pp. 549–559.

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External Links:

Document

Referenced by:

§12.17.

Encodings:

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6: Bibliography O

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

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:

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7: 17.13 Integrals

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17.13.1 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-ax/c;q)}_{\mathrm{\infty }}{(bx/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{(1-q){(q;q)}_{\mathrm{\infty }}{(ab;q)}_{\mathrm{\infty }}cd{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(a;q)}_{\mathrm{\infty }}{(b;q)}_{\mathrm{\infty }}(c+d){(-bc/d;q)}_{\mathrm{\infty }}{(-ad/c;q)}_{\mathrm{\infty }}},$$

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Symbols:

$\int$: integral, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

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17.13.2 $${\int }_{-c}^d\frac{{(-qx/c;q)}_{\mathrm{\infty }}{(qx/d;q)}_{\mathrm{\infty }}}{{(-x{q}^{\alpha }/c;q)}_{\mathrm{\infty }}{(x{q}^{\beta }/d;q)}_{\mathrm{\infty }}}{d}_{q}x=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}\frac{cd}{c+d}\frac{{(-c/d;q)}_{\mathrm{\infty }}{(-d/c;q)}_{\mathrm{\infty }}}{{(-q^{\beta }c/d;q)}_{\mathrm{\infty }}{(-q^{\alpha }d/c;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $q$: complex base and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13 and Ch.17

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial) and $q$: complex base

Referenced by:

Erratum (V1.0.1) for Equation (17.13.3)

Permalink:

http://dlmf.nist.gov/17.13.E3

Encodings:

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Errata (effective with 1.0.1):

Originally the differential was identified incorrectly as a $q$-differential; the correct differential is $dt$.

See also:

Annotations for §17.13, §17.13 and Ch.17

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17.13.4 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-ct{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-ct;q)}_{\mathrm{\infty }}}{d}_{q}t=\frac{{\mathrm{\Gamma }}_q\left(\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right){(-c{q}^{\alpha };q)}_{\mathrm{\infty }}{(-q^{1-\alpha }/c;q)}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_q\left(\alpha +\beta \right){(-c;q)}_{\mathrm{\infty }}{(-q/c;q)}_{\mathrm{\infty }}}.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential and $q$: complex base

Permalink:

http://dlmf.nist.gov/17.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.13, §17.13 and Ch.17

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8: 1.4 Calculus of One Variable

… ►If $f(x)$ is continuous at each point $c\in (a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in C(a,b)$. If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is continuous on the interval $[a,b]$, and we write $f(x)\in C[a,b]$. … ►If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists a point $c\in (a,b)$ such that … ►Let $c\in (a,b)$ and assume that ${\int }_a^{c-ϵ}f(x)dx$ and ${\int }_{c+ϵ}^{b}f(x)dx$ exist when , but not necessarily when $ϵ=0$. … ►If $f(x)\in C^{n+1}[a,b]$, then …

9: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

… ►where $ef=abcq$. … ►For continued-fraction representations of a ratio of ${}_3{}^{}\varphi _2^{}$ functions, see Cuyt et al. (2008, pp. 399–400). … ►where $\lambda =-c{(ab/q)}^{\frac{1}{2}}$. … ►where $a^{2}q=bcde{q}^{-n}$. … ►where $q{a}^2=bcdef$. …

10: Bibliography J

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Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:

A second edition was published in 2010.

External Links:

ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt

Referenced by:

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).

Encodings:

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Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§12.10(vi).

Encodings:

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Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

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External Links:

FullText, MathReview (N. N. Abdelmalek), ZentralBlatt

Referenced by:

§3.10(iii).

Encodings:

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.

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External Links:

ISSN 0075-4102, Document, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(i).

Encodings:

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