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21: Publications

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Toward a Revised NBS Handbook of Mathematical Functions, Technical Report NISTIR 6072 (September 1997), National Institute of Standards and Technology.

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From 2D to 3D: Numerical Grid Generation and the Visualization of Complex Surfaces, Proceedings of the 7th International Conference on Numerical Grid Generation in Computational Field Simulations, Whistler, British Columbia, Canada, September 25-28, 2000.

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The NIST Digital Library of Mathematical Functions Project, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 105–119.

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Technical Aspects of the Digital Library of Mathematical Functions, Annals of Mathematics and Artificial Intelligence—Special Issue on Mathematical Knowledge Management, Vol. 38, Nos. 1–3, pp. 121–136.

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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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22: Frank W. J. Olver

… ►In 1989 the conference “Asymptotic and Computational Analysis” was held in Winnipeg, Canada, in honor of Olver’s 65th birthday, with Proceedings published by Marcel Dekker in 1990. … ►

1 Algebraic and Analytic Methods

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23: Bibliography F

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Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.

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External Links:

ISSN 0377-0427, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.77(vi).

Encodings:

BibTeX

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§29.11.

Encodings:

BibTeX

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On the reciprocal modulus relation for elliptic integrals. SIAM J. Math. Anal. 1 (4), pp. 524–526.

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External Links:

Document, MathReview (B. C. Carlson), ZentralBlatt

Referenced by:

§19.7(i).

Encodings:

BibTeX

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Continuous and Discrete Painlevé Equations. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 33–47.

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Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview (F. Pempinelli), ZentralBlatt

Referenced by:

§32.15.

Encodings:

BibTeX

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

Encodings:

BibTeX

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24: 9.2 Differential Equation

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9.2.1 $$\frac{d^{2}w}{{dz}^2}=zw.$$

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Defines:

$w$: ODE solution (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

Source:

Olver (1997b, (1.01), p. 392)

A&S Ref:

10.4.1 (in slightly different form)

Referenced by:

§36.8, (9.10.10), (9.10.20), (9.10.21), (9.10.8), (9.10.9), §9.10(iii), (9.11.2), §9.11(i), §9.11(iv), §9.12(i), §9.17(ii), §9.17(ii), (9.2.16), §9.2(iii), §9.2(vi), (9.8.14), (9.8.16), (9.8.18), (9.8.19)

Permalink:

http://dlmf.nist.gov/9.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(i), §9.2 and Ch.9

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9.2.2 $$w=\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right),\mathrm{Ai}\left(z{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\right).$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $w$: ODE solution

Source:

Olver (1997b, pp. 392–393, 413–414)

Permalink:

http://dlmf.nist.gov/9.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(i), §9.2 and Ch.9

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9.2.4 $${\mathrm{Ai}}^{\prime }\left(0\right)=-\frac{1}{3^{1/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}=-\mathrm{0.25881\hspace{0.33em}94037}\mathrm{\dots },$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.03), p. 392)

A&S Ref:

10.4.5 (with more digits)

Permalink:

http://dlmf.nist.gov/9.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

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9.2.5 $$\mathrm{Bi}\left(0\right)=\frac{1}{3^{1/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)}=\mathrm{0.61492\hspace{0.33em}66274}\mathrm{\dots },$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function and $\mathrm{\Gamma }\left(z\right)$: gamma function

Source:

Olver (1997b, (1.11), p. 393)

A&S Ref:

10.4.4 (in different form)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.2(ii), §9.2 and Ch.9

… ► $W=(1/w)dw/dz$, where $w$ is any nontrivial solution of (9.2.1). …

25: 28.6 Expansions for Small $q$

… ►The coefficients of the power series of $a_{2n}\left(q\right)$, $b_{2n}\left(q\right)$ and also $a_{2n+1}\left(q\right)$, $b_{2n+1}\left(q\right)$ are the same until the terms in $q^{2n-2}$ and $q^{2n}$, respectively. … ►Numerical values of the radii of convergence ${\rho }_n^{(j)}$ of the power series (28.6.1)–(28.6.14) for $n=0,1,\mathrm{\dots },9$ are given in Table 28.6.1. Here $j=1$ for $a_{2n}\left(q\right)$, $j=2$ for $b_{2n+2}\left(q\right)$, and $j=3$ for $a_{2n+1}\left(q\right)$ and $b_{2n+1}\left(q\right)$. (Table 28.6.1 is reproduced from Meixner et al. (1980, §2.4).) … ►where $k$ is the unique root of the equation $2E\left(k\right)=K\left(k\right)$ in the interval $(0,1)$, and $k^{\prime }=\sqrt{1-k^2}$. …

26: 18.39 Applications in the Physical Sciences

… ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …, $k=1$, there might be 3 nodes, rather than Sturm’s 1. … ►and thus replacing $p$ by $n-l-1$ as in Table 18.8.1, line 11, or as in (18.39.33), … ►a) Shizgal (2015), b) Blackmore et al. (1986), c) Gammaitoni et al. (1998), d) Liboff (2003). … ►with initial data $c_0=1$, $c_{-1}=0$, where …

27: Bibliography C

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Analytical solution of a model for complex food webs. Phys. Rev. E 65 (3), pp. (030901–1)–(030901–4).

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External Links:

Document

Referenced by:

§8.24(i).

Encodings:

BibTeX

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Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity. Phys. Rev. A 62 (063610), pp. 1–10.

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External Links:

Document

Referenced by:

§22.19(iii).

Encodings:

BibTeX

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

Encodings:

BibTeX

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Department of Computer Science, University of Victoria, Canada.

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Notes:

Permutations, set and integer partitions, sequences, graphs, and trees.

External Links:

Combinatorial Object Server

Referenced by:

1st item.

Encodings:

BibTeX

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Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99 (1), pp. 106–114.

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External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

4th item, 1st item.

Encodings:

BibTeX

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28: Bibliography M

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A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.

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External Links:

ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(v).

Encodings:

BibTeX

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The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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External Links:

ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

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On the evaluation of Bessel functions. Appl. Comput. Harmon. Anal. 1 (1), pp. 116–135.

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External Links:

ISSN 1063-5203, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(iii).

Encodings:

BibTeX

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Spin Systems, Statistical Mechanics and Painlevé Functions. In Painlevé Transcendents: Their Asymptotics and Physical Applications, D. Levi and P. Winternitz (Eds.), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 278, pp. 377–391.

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Notes:

Proc. NATO Adv. Res. Workshop, Sainte-Adèle, Canada, 1990

External Links:

MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

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Phenomenological equations of state for the quark-gluon plasma. Phys. Rev. D 65 (3), pp. (034009–1)–(034009–10).

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External Links:

Document

Referenced by:

§24.18.

Encodings:

BibTeX

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29: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►The eigenfunction expansions of (1.8.1) and (1.8.2) follow from Cases 1, 2, above. … ►The resolvent set $\rho (T)$ consists of all $z\in ℂ$ such that (i) $z-T$ is injective, (ii) $ℛ(z-T)$ is dense in $V$, (iii) the resolvent ${\left(z-T\right)}^{-1}$ is bounded. … ►By Weyl’s alternative $n_1$ equals either 1 (the limit point case) or 2 (the limit circle case), and similarly for $n_2$. …If $n_1=1$ then there are no nonzero boundary values at $a$; if $n_1=2$ then the above boundary values at $a$ form a two-dimensional class. … ►The materials developed here follow from the extensions of the Sturm–Liouville theory of second order ODEs as developed by Weyl, to include the limit point and limit circle singular cases. …

30: 13.12 Products

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13.12.1 $$M(a,b,z)M(-a,-b,-z)+\frac{a(a-b)z^2}{b^2(1-b^2)}M(1+a,2+b,z)M(1-a,2-b,-z)=1.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.12, §13.25

Permalink:

http://dlmf.nist.gov/13.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.12 and Ch.13

►For generalizations of this quadratic relation see Majima et al. (2000). ►For integral representations, integrals, and series containing products of $M(a,b,z)$ and $U(a,b,z)$ see Erdélyi et al. (1953a, §6.15.3).