Lambert W-function

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11: DLMF Project News

error generating summary

12: 10.16 Relations to Other Functions

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Elementary Functions

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Airy Functions

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Parabolic Cylinder Functions

… ►For the functions $M_{0,\nu }$ and $W_{0,\nu }$ see §13.14(i). … ►

Generalized Hypergeometric Functions

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13: 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.

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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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Closed-form representations of the Lambert $W$ function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.

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External Links:

ISSN 1311-0454, MathReview (Lewis H. Ryder), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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14: 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.

Encodings:

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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.

Encodings:

BibTeX

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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:

BibTeX

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15: Bibliography C

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Numerical evaluation of the Lambert $W$ function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.

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External Links:

ISSN 1053-587X, Document, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§4.44, §4.45(iii).

Encodings:

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On the Lambert $W$ function. Adv. Comput. Math. 5 (4), pp. 329–359.

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External Links:

ISSN 1019-7168, MathReview, ZentralBlatt

Referenced by:

(4.13.10), (4.13.11), (4.13.1_1), (4.13.5_1), §4.13, §4.44, §4.44.

Encodings:

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A sequence of series for the Lambert $W$ function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.

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External Links:

Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(4.13.4_1), (4.13.4_2), (4.13.5_2), §4.13.

Encodings:

BibTeX

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16: Software Index

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	Open Source													With Book						Commercial
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4.48(iv) Lambert $W$-Function	✓					✓			✓			✓	✓								✓	✓	a
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17: Errata

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§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_k\left(x\right)$, $k\in \mathbb{Z}$, for the branches. The original two solutions are identified via $\mathrm{Wp}\left(x\right)=W_0\left(x\right)$ and $\mathrm{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{d^{n}W}{{dz}^n}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

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4.13.11 $$\mathrm{Wm}\left(x\right)=-\eta -\ln \eta -\frac{\ln \eta }{\eta }+\frac{{(\ln \eta )}^2}{2{\eta }^2}-\frac{\ln \eta }{{\eta }^2}+O\left(\frac{{(\ln \eta )}^3}{{\eta }^3}\right)$$

Originally the sign in front of $\frac{{(\ln \eta )}^2}{2{\eta }^2}$ was $-$. The correct sign is $+$.

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In the final line of this subsection, $\mathrm{Wm}\left(n\right)$ was replaced by $\mathrm{Wp}\left(n\right)$ twice, and the wording was changed from “or, equivalently, $N={\mathrm{e}}^{\mathrm{Wm}\left(n\right)}$” to “or, specifically, $N={\mathrm{e}}^{\mathrm{Wp}\left(n\right)}$”.

Reported by Gergő Nemes on 2018-04-09

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18: Bibliography M

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An integral representation for the Lambert $W$ function.

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External Links:

2012.02480

Referenced by:

(4.13.16), §4.13.

Encodings:

BibTeX

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