Lambert W-function

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1: 4.13 Lambert $W$-Function

§4.13 Lambert $W$-Function

►The Lambert $W$-function $W\left(x\right)$ is the solution of the equation … ►We call the solution for which $W\left(x\right)\ge W\left(-1/\mathrm{e}\right)$ the principal branch and denote it by $\mathrm{Wp}\left(x\right)$. … ►Properties include: … ►For integral representations of all branches of the Lambert $W$-function see Kheyfits (2004). …

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: 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 $[-1/\mathrm{e},0)$. …

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: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For bound-state problems only the exponentially decaying solution is required, usually taken to be the Whittaker function $W_{-\eta ,\mathrm{\ell }+\frac{1}{2}}\left(2\rho \right)$. …

7: 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. …

8: 12.19 Tables

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Fox (1960) includes modulus and phase functions for $W(a,x)$ and $W(a,-x)$, and several auxiliary functions for $x^{-1}=0(.005)0.1$, $a=-10(1)10$, 8S.

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9: 4.1 Special Notation

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$k,m,n$	integers.
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10: 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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