§4.13 Lambert $W$-Function

The Lambert $W$-function $W\left(z\right)$ is the solution of the equation

On the $z$-interval $[0,\mathrm{\infty })$ there is one real solution, and it is nonnegative and increasing. On the $z$-interval $(-{\mathrm{e}}^{-1},0)$ there are two real solutions, one increasing and the other decreasing. 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 $W_0\left(z\right)$. See Figure 4.13.1.

The decreasing solution can be identified as $W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$. Other solutions of (4.13.1) are other branches of $W\left(z\right)$. They are denoted by $W_k\left(z\right)$, $k\in \mathbb{Z}$, and have the property

where ${\ln }_k(z)=\ln \left(z\right)+2\pi \mathrm{i}k$. $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. See (4.13.6) and (4.13.9_1). 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. See Figure 4.13.2.

Alternative notations are $\mathrm{Wp}\left(x\right)$ for $W_0\left(x\right)$, $\mathrm{Wm}\left(x\right)$ for $W_{-1}\left(x+0\mathrm{i}\right)$, both previously used in this section, the Wright $\omega$-function $\omega \left(z\right)=W\left({\mathrm{e}}^z\right)$, which is single-valued, satisfies

and has several advantages over the Lambert $W$-function (see Lawrence et al. (2012)), and the tree $T$-function $T\left(z\right)=-W\left(-z\right)$, which is a solution of

Properties include:

in which the $p_n(x)$ are polynomials of degree $n$ with

Explicit representations for the $p_n(x)$ are given in Kalugin and Jeffrey (2011).

where $t\ge 0$ for $W_0$, $t\le 0$ for $W_{\pm 1}$ on the relevant branch cuts,

and

where $g_n$ is defined in §5.11(i). See Jeffrey and Murdoch (2017) for an explicit representation for the $c_n$ in terms of associated Stirling numbers.

where

For the definition of Stirling cycle numbers of the first kind $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$ see (26.13.3). As $\left|z\right|\to \mathrm{\infty }$

where ${\xi }_k=\ln \left(z\right)+2\pi \mathrm{i}k$. For large enough $\left|z\right|$ the series on the right-hand side of (4.13.10) is absolutely convergent to its left-hand side. In the case of $k=0$ and real $z$ the series converges for $z\ge \mathrm{e}$. As $x\to 0-$

where $\eta =\ln \left(-1/x\right)$. For these results and other asymptotic expansions see Corless et al. (1997).

For integrals of $W\left(z\right)$ use the substitution $w=W\left(z\right)$, $z=w{\mathrm{e}}^w$ and $dz=(w+1){\mathrm{e}}^{w}dw$. Examples are

For these and other integral representations of the Lambert $W$-function see Kheyfits (2004), Kalugin et al. (2012) and Mező (2020).

For the foregoing results and further information see Borwein and Corless (1999), Corless et al. (1996), de Bruijn (1961, pp. 25–28), Olver (1997b, pp. 12–13), and Siewert and Burniston (1973).

For a generalization of the Lambert $W$-function connected to the three-body problem see Scott et al. (2006), Scott et al. (2013) and Scott et al. (2014).