# Lambert W-function

(0.006 seconds)

## 1—10 of 18 matching pages

##### 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 $W\left(z\right)\geq W\left(-{\mathrm{e}}^{-1}\right)=-1$ 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 $\mathbb{C}\setminus(-\infty,0]$, 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 $q$
28.8.9 $W_{m}^{\pm}(x)=\frac{e^{\pm 2h\sin x}}{(\cos x)^{m+1}}\begin{cases}\left(\cos% \left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\\ \left(\sin\left(\frac{1}{2}x+\frac{1}{4}\pi\right)\right)^{2m+1},\end{cases}$
##### 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. …
##### 6: 26.7 Set Partitions: Bell Numbers
or, specifically, $N={\mathrm{e}}^{\operatorname{Wp}\left(n\right)}$, with properties of the Lambert function $\operatorname{Wp}\left(n\right)$ given in §4.13. …
##### 7: 4.45 Methods of Computation
###### §4.45(iii) Lambert$W$-Function
For $x\in[-1/e,\infty)$ the principal branch $\operatorname{Wp}\left(x\right)$ can be computed by solving the defining equation $We^{W}=x$ numerically, for example, by Newton’s rule (§3.8(ii)). … Similarly for $\operatorname{Wm}\left(x\right)$ in the interval $[-1/e,0)$. …
##### 8: 4.1 Special Notation
 $k,m,n$ integers. …
##### 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
• D. Jacobs and F. Lambert (1972) On the numerical calculation of polylogarithms. Nordisk Tidskr. Informationsbehandling (BIT) 12 (4), pp. 581–585.
• D. J. Jeffrey and N. Murdoch (2017) 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.