Lambert W-function

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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 … ►We call the increasing solution for which $W\left(z\right)\ge W\left(-{\mathrm{e}}^{-1}\right)=-1$ the principal branch and denote it by $W_0\left(z\right)$. … ►The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. … ►Properties include: …

2: 4.44 Other Applications

… ►For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). ►For an application of the Lambert $W$-function to generalized Gaussian noise see Chapeau-Blondeau and Monir (2002). For other applications of the Lambert $W$-function see Corless et al. (1996).

3: 28.8 Asymptotic Expansions for Large $q$

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28.8.9 $$W_m^{\pm }(x)=\frac{{\mathrm{e}}^{\pm 2h\sin x}}{{(\cos x)}^{m+1}}\{\begin{array}{cc}{\left(\cos \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \\ {\left(\sin \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \end{array}$$

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Defines:

$W_m^{\pm }$: function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.13 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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4: 27.7 Lambert Series as Generating Functions

§27.7 Lambert Series as Generating Functions

►Lambert series have the form …

5: 4.48 Software

… ►Links to research literature for the Lambert $W$-function and for test software are included also. … ►

§4.48(iv) Lambert $W$-Function

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6: 26.7 Set Partitions: Bell Numbers

… ►or, specifically, $N={\mathrm{e}}^{\mathrm{Wp}\left(n\right)}$, with properties of the Lambert function $\mathrm{Wp}\left(n\right)$ given in §4.13. …

7: 4.45 Methods of Computation

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§4.45(iii) Lambert $W$-Function

►For $x\in [-1/\mathrm{e},\mathrm{\infty })$ the principal branch $\mathrm{Wp}\left(x\right)$ can be computed by solving the defining equation $W{\mathrm{e}}^W=x$ numerically, for example, by Newton’s rule (§3.8(ii)). … ►Similarly for $\mathrm{Wm}\left(x\right)$ in the interval $[-1/\mathrm{e},0)$. …

8: 4.1 Special Notation

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$k,m,n$	integers.
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9: 25.18 Methods of Computation

… ►For dilogarithms and polylogarithms see Jacobs and Lambert (1972), Osácar et al. (1995), Spanier and Oldham (1987, pp. 231–232), and Zudilin (2007). …

10: Bibliography J

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On the numerical calculation of polylogarithms. Nordisk Tidskr. Informationsbehandling (BIT) 12 (4), pp. 581–585.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Stirling Numbers, Lambert W and the Gamma Function. In Mathematical Aspects of Computer and Information Sciences, J. Blömer, I. S. Kotsireas, T. Kutsia, and D. E. Simos (Eds.), Cham, pp. 275–279.

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External Links:

ISBN 978-3-319-72453-9, ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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