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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: Software Index

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	Open Source													With Book						Commercial
…
4.48(iv) Lambert $W$-Function	✓					✓			✓			✓	✓								✓	✓	a
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7.25(iii) $\mathrm{erf}z$, $\mathrm{erfc}z$, $w\left(z\right)$, $z\in ℂ$	✓								✓			a	✓	✓							✓	✓	✓
…
13.32(ii) $M(a,b,x)$, $U(a,b,x)$, $𝐌(a,b,x)$, $M_{\kappa ,\mu }\left(x\right)$, $W_{\kappa ,\mu }\left(x\right)$, $x,a,b\in ℝ$	✓		✓	✓	✓	✓						✓	✓	✓			✓		✓		✓	✓	a
13.32(iii) $M(a,b,z)$, $U(a,b,z)$, $𝐌(a,b,z)$, $M_{\kappa ,\mu }\left(z\right)$, $W_{\kappa ,\mu }\left(z\right)$, $z,a,b\in ℂ$	✓											✓	✓	✓					✓		✓	✓	a
…

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These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

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3: 27.20 Methods of Computation: Other Number-Theoretic Functions

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

4: Bibliography E

… ►

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

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External Links:

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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External Links:

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

Referenced by:

§10.21(iv).

Encodings:

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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).

Encodings:

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

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External Links:

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

Referenced by:

§18.34(ii).

Encodings:

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

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§10.37.

Encodings:

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5: 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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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.45(iii).

Encodings:

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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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External Links:

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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External Links:

ISSN 0893-9659, MathReview, ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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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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External Links:

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

Referenced by:

§25.18(i).

Encodings:

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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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External Links:

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

Referenced by:

§2.11(v).

Encodings:

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6: 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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External Links:

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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External Links:

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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7: 23.10 Addition Theorems and Other Identities

… ►If $u+v+w=0$, then ►

23.10.5

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): 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

Encodings:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

See also:

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

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8: 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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External Links:

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

Referenced by:

§4.13, §4.13.

Encodings:

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

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External Links:

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

Referenced by:

§21.9.

Encodings:

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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).

Encodings:

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9: 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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External Links:

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

Referenced by:

§4.13.

Encodings:

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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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External Links:

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

Referenced by:

§4.13.

Encodings:

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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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External Links:

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

Referenced by:

§5.6(i).

Encodings:

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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.

Encodings:

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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).

Encodings:

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10: 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.

Encodings:

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

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External Links:

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

Referenced by:

§19.9(i).

Encodings:

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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.

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).

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