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11: Bibliography G

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W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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External Links:: ISSN 0001-0782, Document
Referenced by:: §33.26(ii).
Encodings:: BibTeX

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W. Gautschi (1969) Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Notes:: Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iii).
Encodings:: BibTeX

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A. Gil, J. Segura, and N. M. Temme (2002c) 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:: BibTeX

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H. W. Gould (1960) Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:: FullText, MathReview (L. Carlitz), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

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V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:: In Russian. English translation: Differential Equations 11(11), pp. 285–287
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(iii), §32.7(vi).
Encodings:: BibTeX

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12: Bibliography B

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R. Barakat (1961) Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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External Links:: FullText, MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §8.7.
Encodings:: BibTeX

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B. C. Berndt, S. Bhargava, and F. G. Garvan (1995) Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:: ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt
Referenced by:: §15.8(v), §20.11(iii).
Encodings:: BibTeX

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F. Bethuel (1998) Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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External Links:: ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §17.17.
Encodings:: BibTeX

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A. Bhattacharyya and L. Shafai (1988) Theoretical and experimental investigation of the elliptical annual ring antenna. IEEE Trans. Antennas and Propagation 36 (11), pp. 1526–1530.

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External Links:: Document
Referenced by:: 5th item.
Encodings:: BibTeX

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R. L. Bishop (1981) Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

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External Links:: ISSN 0035-9149, Document, MathReview
Referenced by:: Sidebar 9.SB1.
Encodings:: BibTeX

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13: Bibliography

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M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:: MathReview (J. G. van der Corput), ZentralBlatt
Referenced by:: §30.9(iii).
Encodings:: BibTeX

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S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:: ISSN 0024-1318, MathReview (James M. Horner)
Referenced by:: §13.9(i).
Encodings:: BibTeX

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V. I. Arnol’d (1974) Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:: Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.
External Links:: Document, MathReview (G. R. Belickii), ZentralBlatt
Referenced by:: §36.2(i).
Encodings:: BibTeX

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V. I. Arnol’d (1975) Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:: In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.
External Links:: Document, MathReview (D. Siersma), ZentralBlatt
Referenced by:: §36.2(i), Ch.36.
Encodings:: BibTeX

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R. Askey (1980) Some basic hypergeometric extensions of integrals of Selberg and Andrews. SIAM J. Math. Anal. 11 (6), pp. 938–951.

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External Links:: ISSN 0036-1410, Document, MathReview (W. A. Al-Salam), ZentralBlatt
Referenced by:: §17.13, §17.13.
Encodings:: BibTeX

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Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$ , $-x^{-2}\operatorname{Cin}\left(x\right)$ , $x^{-1}\operatorname{Ein}\left(x\right)$ , $-x^{-1}\operatorname{Ein}\left(-x\right)$ , $x=0(.01)0.5$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=0.5(.01)2$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x=2(.1)10$ ; $x\mathrm{f}\left(x\right)$ , $x^{2}\mathrm{g}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x^{-1}=0(.005)0.1$ ; $\operatorname{Si}\left(\pi x\right)$ , $\operatorname{Cin}\left(\pi x\right)$ , $x=0(.1)10$ . Accuracy varies but is within the range 8S–11S.

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Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

15: 8.26 Tables

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Zhang and Jin (1996, Table 3.8) tabulates $\gamma\left(a,x\right)$ for $a=0.5,1,3,5,10,25,50,100$ , $x=0(.1)1(1)3,5(5)30,50,100$ to 8D or 8S.

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Pearson (1968) tabulates $I_{x}\left(a,b\right)$ for $x=0.01(.01)1$ , $a,b=0.5(.5)11(1)50$ , with $b\leq a$ , to 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_{n}\left(x\right)$ for $n=1,2,3,5,10,15,20$ , $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

16: 30 Spheroidal Wave Functions

Chapter 30 Spheroidal Wave Functions

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17: Bibliography K

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G. A. Kalugin, D. J. Jeffrey, and R. M. Corless (2012) Bernstein, Pick, Poisson and related integral expressions for Lambert

W

. Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:: ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt
Referenced by:: §4.13.
Encodings:: BibTeX

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E. L. Kaplan (1948) Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:: MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §19.12.
Encodings:: BibTeX

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R. P. Kerr (1963) Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.

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External Links:: Document, MathReview (C. Gilbert), ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

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K. S. Kölbig (1968) Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §25.21(v).
Encodings:: BibTeX

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T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §15.14, §15.14.
Encodings:: BibTeX

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18: 34.6 Definition: $\mathit{9j}$ Symbol

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34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

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Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

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34.6.2

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{j}(-1)^{2j}(2j+1)\begin{Bmatrix}j_{11}% &j_{21}&j_{31}\\ j_{32}&j_{33}&j\end{Bmatrix}\begin{Bmatrix}j_{12}&j_{22}&j_{32}\\ j_{21}&j&j_{23}\end{Bmatrix}\begin{Bmatrix}j_{13}&j_{23}&j_{33}\\ j&j_{11}&j_{12}\end{Bmatrix}.

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Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Referenced by:: §34.13
Permalink:: http://dlmf.nist.gov/34.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

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19: 31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

20: 26.12 Plane Partitions

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Table 26.12.1: Plane partitions.

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$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$	$n$	$\operatorname{pp}\left(n\right)$
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11	859	28	24 83234	45	17740 79109
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13	2485	30	56 68963	47	36379 93036
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26.12.26

\operatorname{pp}\left(n\right)\sim\frac{\left(\zeta\left(3\right)\right)^{7/3% 6}}{2^{11/36}(3\pi)^{1/2}n^{25/36}}\exp\left(3\left(\zeta\left(3\right)\right)% ^{1/3}\left(\tfrac{1}{2}n\right)^{2/3}+\zeta'\left(-1\right)\right),

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Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\sim$ : asymptotic equality, $\exp\NVar{z}$ : exponential function, $\operatorname{pp}\left(\NVar{n}\right)$ : number of plane partitions of $n$ , $n$ : nonnegative integer and $\pi$ : plane partition
Referenced by:: §26.12(iv), Erratum (V1.0.5) for Equation (26.12.26)
Permalink:: http://dlmf.nist.gov/26.12.E26
Encodings:: pMML, png, TeX
Errata (effective with 1.0.5):: The original statement of this equation, $\operatorname{pp}\left(n\right)\sim\left(\frac{\zeta\left(3\right)}{2^{11}n^{2% 5}}\right)^{1/36}\*\exp\left(3\left(\frac{\zeta\left(3\right)n^{2}}{4}\right)^% {1/3}+\zeta'\left(-1\right)\right)$ , has been corrected.
Reported 2011-09-05 by Suresh Govindarajan
See also:: Annotations for §26.12(iv), §26.12 and Ch.26

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