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11: Bibliography E

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C. Eckart (1930) The penetration of a potential barrier by electrons. Phys. Rev. 35 (11), pp. 1303–1309.

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External Links:: Document, ZentralBlatt
Referenced by:: §15.18.
Encodings:: BibTeX

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A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953a) Higher Transcendental Functions. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:: Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1385, v. 41 (1983), no. 164, p. 778, v. 30 (1976), no. 135, p. 675, v. 25 (1971), no. 115, p. 635, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 112, p. 999, v. 24 (1970), no. 110, p. 504, v. 17 (1963), no. 84, p. 485.
External Links:: MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §13.1, §13.10(ii), §13.10(iv), §13.12, §13.13(i), §13.16(ii), §13.16(iii), §13.23(i), §13.23(ii), §13.25, §13.4(i), §13.4(ii), §13.9(i), §13.9(ii), Ch.13, §14.1, §14.10, §14.12(ii), §14.12(i), §14.12(ii), §14.13, §14.13, §14.17(ii), §14.17(iv), §14.18(ii), §14.18(iii), §14.18(iv), §14.19(i), §14.19(ii), §14.19(iii), §14.19(iv), §14.20(v), §14.21(iii), §14.25, §14.28(i), §14.28(ii), §14.3(iii), §14.3(iii), §14.3(iv), §14.5(iii), §14.5(iv), §14.6(i), §14.6(ii), §14.7(iv), §14.8(i), §14.8(ii), Ch.14, §15.16, §15.4(iii), §15.5(i), §15.6, §15.8(v), §15.8(ii), §15.8(v), §15.9(iii), Ch.15, §16.12, §16.12, §16.13, §16.14(i), §16.14(ii), (16.15.3), §16.15, §16.16(i), §16.16(ii), §16.18, §16.20, §16.20, §16.21, §16.4(ii), §16.5, §16.6, Ch.16, §19.21(i), §19.28, §19.5, §24.16(iii), §24.5(i), §24.7(i), §24.7(ii), §24.8(i), Ch.24, (25.11.12), (25.11.18), (25.11.29), (25.11.30), (25.12.12), (25.12.13), (25.14.1), (25.14.4), (25.14.5), (25.14.6), §25.14(i), §25.14(i), §25.14(ii), §25.14(ii), §25.14, (25.5.1), (25.5.10), (25.5.11), (25.5.20), (25.5.21), (25.5.3), (25.5.8), (25.5.9), §25.5(i), §25.5(iii), §25.5, (25.8.5), (25.8.6), §25.8, Ch.25, §5.7(i), Ch.5, Notations P, Notations P, Notations Q.
Encodings:: BibTeX

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A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1953b) Higher Transcendental Functions. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:: Reprinted by Robert E. Krieger Publishing Co. Inc., 1981. Table errata: Math. Comp. v. 41 (1983), no. 164, p. 778, v. 36 (1981), no. 153, p. 315, v. 30 (1976), no. 135, p. 675, v. 29 (1975), no. 130, p. 670, v. 26 (1972), no. 118, p. 598, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239, v. 17 (1963), no. 84, p. 485.
External Links:: MathReview (E. T. Copson), ZentralBlatt
Referenced by:: §10.22(iv), §10.22(vi), §10.23(iii), §10.23(iii), §10.23(iv), §10.32(i), §10.32(iii), §10.32(iv), §10.43(iv), §10.43(vi), §10.44(iv), §10.60(iv), §10.9(i), §10.9(iii), §10.9(iv), §11.10(vi), §11.4(i), Ch.11, §12.12, §12.12, §12.13(i), §12.5(iv), Ch.12, §18.16(vi), (18.17.21_1), §18.17(iv), §18.17(i), §18.17(viii), (18.35.1), (18.35.2), (18.35.4_5), §18.35(i), (18.5.11_1), (18.5.11_3), §18.5(ii), §18.9(iii), §19.1, §19.10(i), §19.14(ii), (19.25.40), §19.25(vi), §19.5, §19.7(ii), §19.9(i), Ch.19, §20.9(i), §22.15(ii), §23.6(iv), Ch.23, §8.13(i), §8.14, §8.3(i), §8.4, §8.6(iii), §8.7, Ch.8, Notations P, Notations P.
Encodings:: BibTeX

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F. H. L. Essler, H. Frahm, A. R. Its, and V. E. Korepin (1996) Painlevé transcendent describes quantum correlation function of the

X X Z

antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.

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External Links:: ISSN 0305-4470, Document, MathReview (Estelle L. Basor), ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

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L. Euler (1768) Institutiones Calculi Integralis. Opera Omnia (1), Vol. 11, pp. 110–113.

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Referenced by:: §25.12(i).
Encodings:: BibTeX

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Slater (1960) tabulates $M\left(a,b,x\right)$ for $a=-1(.1)1$ , $b=0.1(.1)1$ , and $x=0.1(.1)10$ , 7–9S; $M\left(a,b,1\right)$ for $a=-11(.2)2$ and $b=-4(.2)1$ , 7D; the smallest positive $x$ -zero of $M\left(a,b,x\right)$ for $a=-4(.1){-}0.1$ and $b=0.1(.1)2.5$ , 7D.

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Zhang and Jin (1996, pp. 411–423) tabulates $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$ for $a=-5(.5)5$ , $b=0.5(.5)5$ , and $x=0.1,1,5,10,20,30$ , 8S (for $M\left(a,b,x\right)$ ) and 7S (for $U\left(a,b,x\right)$ ).

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13: Hans Volkmer

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29 Lamé Functions

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30 Spheroidal Wave Functions

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14: Bibliography L

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D. Le (1985) An efficient derivative-free method for solving nonlinear equations. ACM Trans. Math. Software 11 (3), pp. 250–262.

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Notes:: Corrigendum, ibid. 15(3) (1989), 287
External Links:: ISSN 0098-3500, Document, MathReview (T. Feagin), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

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A. Leitner and J. Meixner (1960) Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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External Links:: ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt
Referenced by:: §30.12.
Encodings:: BibTeX

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H. Lotsch and M. Gray (1964) Algorithm 244: Fresnel integrals. Comm. ACM 7 (11), pp. 660–661.

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Notes:: Includes Algol program, maximum accuracy 12D.
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iv).
Encodings:: BibTeX

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N. A. Lukaševič (1967b) On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).

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Notes:: In Russian. English translation: Differential Equations 3(11), pp. 994–999
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).
Encodings:: BibTeX

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Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.

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External Links:: MathReview (F. Gotze), ZentralBlatt
Referenced by:: §13.31(iii).
Encodings:: BibTeX

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15: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of

\frac{1}{12}\pi

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$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
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$11\pi/12$	$\frac{1}{4}\sqrt{2}(\sqrt{3}-1)$	$-\frac{1}{4}\sqrt{2}(\sqrt{3}+1)$	$-(2-\sqrt{3})$	$\sqrt{2}(\sqrt{3}+1)$	$-\sqrt{2}(\sqrt{3}-1)$	$-(2+\sqrt{3})$
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16: 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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17: 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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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: Bibliography O

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K. Okamoto (1987c) Studies on the Painlevé equations. IV. Third Painlevé equation

P_{{\rm III}}

. Funkcial. Ekvac. 30 (2-3), pp. 305–332.

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External Links:: ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(iii), §32.2(iii), §32.6(iv), §32.7(viii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (1998a) 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:: BibTeX

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F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.

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Notes:: See also Olver (1997b)
External Links:: Document, MathReview (L. Berg), ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

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S. Olver (2011) Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

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External Links:: ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §32.17, §32.17.
Encodings:: BibTeX

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H. Oser (1960) Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

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Notes:: Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.
External Links:: ISSN 0001-0782, Document
Referenced by:: §10.77(ii).
Encodings:: BibTeX

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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
12	1479	29	37 59612	46	25435 35902
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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