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1: 34.9 Graphical Method

… ►For an account of this method see Brink and Satchler (1993, Chapter VII). For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

2: Bibliography

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Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.

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External Links:

ISSN 0025-584X, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.27(vii).

Encodings:

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Solid State Physics. Holt, Rinehart and Winston, New York.

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External Links:

ISBN 0-03-083993-9

Referenced by:

§1.18(vii), §4.44, §8.22(ii).

Encodings:

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Orthogonal Polynomials and Special Functions. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 21, Society for Industrial and Applied Mathematics, Philadelphia, PA.

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External Links:

ISBN 0-89871-018-9, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(ii), §18.18(vii), Profile Richard A. Askey.

Encodings:

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Continuous $q$-Hermite Polynomials when $q>1$ . In $q$-series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.

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External Links:

MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

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Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

Encodings:

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3: 24.2 Definitions and Generating Functions

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24.2.3 $$\frac{t{\mathrm{e}}^{xt}}{{\mathrm{e}}^t-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_n\left(x\right)\frac{t^n}{n!},$$ .

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex and $x$: real or complex

A&S Ref:

23.1.1

Referenced by:

§18.2(xii), §24.4(iv), §24.4(vii)

Permalink:

http://dlmf.nist.gov/24.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(i), §24.2 and Ch.24

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24.2.8 $$\frac{2{\mathrm{e}}^{xt}}{{\mathrm{e}}^t+1}=\sum _{n=0}^{\mathrm{\infty }}{E}_n\left(x\right)\frac{t^n}{n!},$$ ,

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex and $x$: real or complex

A&S Ref:

23.1.1

Referenced by:

§18.2(xii), §24.4(iv), §24.4(vii)

Permalink:

http://dlmf.nist.gov/24.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(ii), §24.2 and Ch.24

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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$30$	$\mathrm{861\hspace{0.33em}58412\hspace{0.33em}76005}$	$14322$	$\mathrm{6.01580\hspace{0.33em}8739}$	$\mathrm{\times 10⁸}$
…

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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$30$	$-\mathrm{44\hspace{0.33em}15438\hspace{0.33em}93249\hspace{0.33em}02310\hspace{0.33em}45536\hspace{0.33em}82821}$
…

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Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

► ►►►

	$k$
…
$11$	$0$	$\frac{5}{6}$	$0$	$-\frac{11}{2}$	$0$	$11$	$0$	$-11$	$0$	$\frac{55}{6}$	$-\frac{11}{2}$	$1$
…

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…

4: Bibliography E

… ►

Angular Momentum in Quantum Mechanics. 3rd printing, with corrections, 2nd edition, Princeton University Press, Princeton, NJ.

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Notes:

Reprinted in 1996.

External Links:

ISBN 0-691-02589-4, MathReview, ZentralBlatt

Referenced by:

§14.30(i), §14.30(ii), §14.30(iii), §14.30(iv), §14.30(iv), §14.31(iii), (34.1.1), §34.1, §34.1, §34.13, §34.2, §34.3(i), §34.3(i), §34.3(ii), §34.3(iii), §34.3(iv), §34.3(vii), §34.3(vii), §34.4, §34.5(i), §34.5(ii), §34.5(iii), §34.5(iv), §34.5(vi), §34.6, §34.7(i), §34.7(ii), §34.7(iv), §34.7(vi), Ch.34, Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1.

Encodings:

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The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

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External Links:

ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

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The asymptotic behaviour of the inhomogeneous Airy function $\mathrm{Hi}(z)$ . Math. Chronicle 12, pp. 99–104.

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External Links:

ISSN 0581-1155, MathReview (Ram Kishore Saxena), ZentralBlatt

Referenced by:

(9.12.24), §9.12(vii).

Encodings:

BibTeX

5: Bibliography L

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The inhomogeneous Airy functions, $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ . J. Chem. Phys. 72 (1), pp. 332–336.

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External Links:

ISSN 0021-9606, Document, MathReview (V. L. Deshpande)

Referenced by:

(9.12.21), §9.12(vii), §9.17(iii).

Encodings:

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Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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… ►

Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.

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External Links:

Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

Encodings:

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Bessel transforms and rational extrapolation. Numer. Math. 47 (1), pp. 1–14.

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External Links:

ISSN 0029-599X, Document, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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Integrating some infinite oscillating tails. J. Comput. Appl. Math. 12/13, pp. 109–117.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

BibTeX

6: Bibliography C

… ►

A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

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External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

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External Links:

ISSN 0010-4655, Document, MathReview

Referenced by:

§10.74(vii).

Encodings:

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Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Mohammad Ahmad), ZentralBlatt

Referenced by:

§20.11(v), §20.7(i), §20.7(i), §20.7(ii), §20.7(ii), §20.7(ii), §20.7(iii), §20.7(iii), §20.7(vii), §20.7(vii), Profile Bille C. Carlson.

Encodings:

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Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

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External Links:

Document

Referenced by:

§10.74(vii).

Encodings:

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Computation of Hankel transforms. SIAM Rev. 14 (2), pp. 278–285.

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External Links:

ISSN 1095-7200, Document, MathReview (R. D. Riess), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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7: 9.18 Tables

… ►

Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

… ►

National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

… ►

Gil et al. (2003c) tabulates the only positive zero of ${\mathrm{Gi}}^{\prime }\left(z\right)$, the first 10 negative real zeros of $\mathrm{Gi}\left(z\right)$ and ${\mathrm{Gi}}^{\prime }\left(z\right)$, and the first 10 complex zeros of $\mathrm{Gi}\left(z\right)$, ${\mathrm{Gi}}^{\prime }\left(z\right)$, $\mathrm{Hi}\left(z\right)$, and ${\mathrm{Hi}}^{\prime }\left(z\right)$. Precision is 11 or 12S.

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§9.18(vii) Generalized Airy Functions

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8: Bibliography G

… ►

On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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External Links:

ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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The triplets of helium. Philos. Trans. Roy. Soc. London Ser. A 228, pp. 151–196.

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External Links:

Document, ZentralBlatt (Möglich, Dr. F.)

Referenced by:

§34.3(vii).

Encodings:

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Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.

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External Links:

ISSN 1017-1398, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§9.17(iii).

Encodings:

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… ►

Matrix Computations. 3rd edition, Johns Hopkins University Press, Baltimore, MD.

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External Links:

ISBN 0-8018-5413-X; 0-8018-5414-8, MathReview, ZentralBlatt

Referenced by:

§3.2(i), §3.2(i), §3.2(ii), §3.2(vii).

Encodings:

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Constructing wavefunctions for nonlocal potentials. J. Chem. Phys. 52, pp. 6211–6217.

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External Links:

Document, MathReview (D. B. Fairlie)

Referenced by:

(9.12.22), (9.12.23), §9.12(vii), §9.17(iii).

Encodings:

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…

9: Bibliography B

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The generating function of Jacobi polynomials. J. London Math. Soc. 13, pp. 8–12.

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External Links:

ISSN 0024-6107; 1469-7750/e, Document, ZentralBlatt

Referenced by:

§18.18(vii).

Encodings:

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Numerical evaluation of the zero-order Hankel transform using Filon quadrature philosophy. Appl. Math. Lett. 9 (5), pp. 21–26.

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External Links:

ISSN 0893-9659, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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KLEIN: Coulomb functions for real $\lambda$ and positive energy to high accuracy. Comput. Phys. Comm. 24 (2), pp. 141–159.

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Notes:

Double-precision Fortran code for positive energies.

External Links:

Document, Code

Referenced by:

5th item, §33.23(vii), 1st item.

Encodings:

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On numerical evaluation of integrals involving Bessel functions. Apl. Mat. 31 (5), pp. 396–410.

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External Links:

ISSN 0373-6725, MathReview (G. A. Evans), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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The determination of phases and amplitudes of wave functions. Proc. Phys. Soc. 81 (3), pp. 442–452.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

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10: Bibliography O

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Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{\mathrm{VI}}$ . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

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External Links:

ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).

Encodings:

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A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

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Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

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External Links:

Document, MathReview (F. Oberhettinger), ZentralBlatt

Referenced by:

§10.19(iii), §10.21(vii).

Encodings:

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Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.

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External Links:

MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§12.10(i), §12.10(ii), §12.10(iii), §12.10(iv), §12.10(v), §12.10(vii), §12.10(vii), §12.11(iii), §12.11(iii), §12.14(ix), §12.14(ix), §12.14(ix), §12.14(xi), Ch.12, §18.16(v).

Encodings:

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Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:

Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania

External Links:

MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))

Referenced by:

§3.8(vii).

Encodings:

BibTeX

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