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1: 4.13 Lambert $W$-Function

§4.13 Lambert $W$-Function

►The Lambert $W$-function $W\left(z\right)$ is the solution of the equation … ►Other solutions of (4.13.1) are other branches of $W\left(z\right)$. … ►Properties include: … ►For integrals of $W\left(z\right)$ use the substitution $w=W\left(z\right)$, $z=w{\mathrm{e}}^w$ and $dz=(w+1){\mathrm{e}}^{w}dw$. …

2: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►See Calkin et al. (2007), and Lehmer (1941, pp. 5–83). …

3: Bibliography E

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Properties of a Raman atom-laser output coupler. J. Phys. B 32 (12), pp. 2935–2950.

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FullText

Referenced by:

§12.17.

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Some recent results on the zeros of Bessel functions and orthogonal polynomials. J. Comput. Appl. Math. 133 (1-2), pp. 65–83.

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ISSN 0377-0427, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.21(iv).

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Waring’s problem. Amer. Math. Monthly 78 (1), pp. 10–36.

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External Links:

Document, FullText, MathReview (J. W. S. Cassels), ZentralBlatt

Referenced by:

§27.13(iii), §27.13(iii).

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Real orthogonalizing weights for Bessel polynomials. J. Comput. Appl. Math. 49 (1-3), pp. 51–57.

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ISSN 0377-0427, Document, MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§18.34(ii).

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On an integral inequality. Proc. Roy. Soc. London Ser. A 357, pp. 271–288.

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ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§10.37.

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4: Bibliography B

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Real values of the $W$-function. ACM Trans. Math. Software 21 (2), pp. 161–171.

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ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.45(iii).

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

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Bernoulli numbers and polynomials of arbitrary complex indices. Appl. Math. Lett. 5 (6), pp. 83–88.

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ISSN 0893-9659, MathReview, ZentralBlatt

Referenced by:

§24.16(iii).

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Riemann zeta function: Rapidly converging series and integral representations. Appl. Math. Lett. 5 (2), pp. 83–88.

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ISSN 0893-9659, Document, MathReview (Jacek Fabrykowski), ZentralBlatt

Referenced by:

§25.18(i).

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The Borel transform and its use in the summation of asymptotic expansions. Stud. Appl. Math. 105 (2), pp. 83–113.

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ISSN 0022-2526, Document, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.11(v).

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5: Bibliography I

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Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.

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MathReview (H. Bateman), ZentralBlatt

Referenced by:

§29.1, §29.17(ii), §29.3(iii), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations E, Notations E, Notations E, Notations E.

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$q$-Series from a Contemporary Perspective. Contemporary Mathematics, Vol. 254, American Mathematical Society, Providence, RI.

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ISBN 0-8218-1150-9, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

Profile Mourad E.H. Ismail.

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The Riemann Zeta-Function. A Wiley-Interscience Publication, John Wiley & Sons Inc., New York.

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Notes:

Reprinted by Dover, 2003.

External Links:

ISBN 0-471-80634-X, MathReview (S. W. Graham), ZentralBlatt

Referenced by:

(25.2.4), (25.2.5), §25.2(i), §25.2(ii), §25.2, Ch.25.

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6: 23.10 Addition Theorems and Other Identities

… ►If $u+v+w=0$, then ►

23.10.5

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$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $det$: determinant, $𝕃$: lattice and $w=-u-v$

Referenced by:

§22.8(iii)

Permalink:

http://dlmf.nist.gov/23.10.E5

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See also:

Annotations for §23.10(i), §23.10 and Ch.23

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23.10.6 $${\left(\zeta \left(u\right)+\zeta \left(v\right)+\zeta \left(w\right)\right)}^2+{\zeta }^{\prime }\left(u\right)+{\zeta }^{\prime }\left(v\right)+{\zeta }^{\prime }\left(w\right)=0.$$

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Symbols:

$\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $𝕃$: lattice and $w=-u-v$

Permalink:

http://dlmf.nist.gov/23.10.E6

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pMML, png, TeX

See also:

Annotations for §23.10(i), §23.10 and Ch.23

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7: Bibliography S

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Numerics of the generalized Lambert $W$ function. ACM Commun. Comput. Algebra 48 (2), pp. 42–56.

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External Links:

FullText, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, §4.13.

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Asymptotic series of generalized Lambert $W$ function. ACM Commun. Comput. Algebra 47 (3), pp. 75–83.

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External Links:

FullText, ZentralBlatt

Referenced by:

§4.13, §4.13.

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General relativity and quantum mechanics: towards a generalization of the Lambert $W$ function: a generalization of the Lambert $W$ function. Appl. Algebra Engrg. Comm. Comput. 17 (1), pp. 41–47.

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ISSN 0938-1279, Document, FullText, MathReview (E. Capelas de Oliveira), ZentralBlatt

Referenced by:

§4.13, §4.13.

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

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Text-book on Spherical Astronomy. Fifth edition, Cambridge University Press, Cambridge.

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External Links:

MathReview

Referenced by:

§4.42(iii).

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8: Bibliography K

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Unimodal sequences show that Lambert $W$ is Bernstein. C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.

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ISSN 0706-1994, MathReview (Fana Tangara), ZentralBlatt

Referenced by:

§4.13.

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Bernstein, Pick, Poisson and related integral expressions for Lambert $W$ . Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt

Referenced by:

§4.13.

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Some extensions of W. Gautschi’s inequalities for the gamma function. Math. Comp. 41 (164), pp. 607–611.

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ISSN 0025-5718, FullText, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§5.6(i).

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Fourier Analysis. 2nd edition, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-38991-7, MathReview, ZentralBlatt

Referenced by:

§1.15(iii), §20.13.

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The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.

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External Links:

MathReview (G. L. Lamb, Jr.), ZentralBlatt

Referenced by:

§22.19(iii).

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9: Bibliography

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Algorithm 724: Program to calculate F-percentiles. ACM Trans. Math. Software 19 (4), pp. 481–483.

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Notes:

Double-precision Fortran, maximum accuracy 13D.

External Links:

ISSN 0098-3500, Document, GAMS (module 724; package TOMS), ZentralBlatt

Referenced by:

1st item.

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

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

Referenced by:

§18.27(vii).

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Functional inequalities for complete elliptic integrals and their ratios. SIAM J. Math. Anal. 21 (2), pp. 536–549.

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ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§19.9(i).

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Tables of Lamé Polynomials. Pergamon Press, The Macmillan Co., New York.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§29.15(ii), 2nd item.

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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).

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10: Bibliography V

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Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.

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ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.60(iii), §10.60(iii).

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Dritter Beweis der die unvollständige Gammafunktion betreffenden Lochsschen Ungleichungen. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 192 (1-3), pp. 83–91 (German).

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ISSN 0723-9319, MathReview (Peter Schroth), ZentralBlatt

Referenced by:

§8.10.

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