# Β§4.13 Lambert $W$-Function

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

 4.13.1 $W{\mathrm{e}}^{W}=z.$ β Defines: $W\left(\NVar{z}\right)$: Lambert $W$-function Symbols: $\mathrm{e}$: base of natural logarithm and $z$: complex variable Referenced by: Β§4.13 Permalink: http://dlmf.nist.gov/4.13.E1 Encodings: TeX, pMML, png See also: Annotations for Β§4.13 and Ch.4

On the $z$-interval $[0,\infty)$ 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 $W\left(z\right)\geq W\left(-{\mathrm{e}}^{-1}\right)=-1$ 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

 4.13.1_1 $W_{k}\left(z\right)={\rm ln}_{k}(z)-\ln\left({\rm ln}_{k}(z)\right)+o\left(1% \right),$ $\left|z\right|\to\infty$, β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $o\left(\NVar{x}\right)$: order less than, $\ln\NVar{z}$: principal branch of logarithm function, $k$: integer and $z$: complex variable Notes: See Corless et al. (1996, Β§4). Permalink: http://dlmf.nist.gov/4.13.E1_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4

where ${\rm ln}_{k}(z)=\ln\left(z\right)+2\pi\mathrm{i}k$. $W_{0}\left(z\right)$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,-{\mathrm{e}}^{-1}]$, 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 $\mathbb{C}\setminus(-\infty,0]$, 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. See Figure 4.13.2.

Alternative notations are $\operatorname{Wp}\left(x\right)$ for $W_{0}\left(x\right)$, $\operatorname{Wm}\left(x\right)$ 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

 4.13.1_2 $\omega\left(z\right)+\ln\left(\omega\left(z\right)\right)=z,$ β Symbols: $\omega\left(\NVar{z}\right)$: Wright $\omega$-function, $\ln\NVar{z}$: principal branch of logarithm function and $z$: complex variable Referenced by: (4.13.3), Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E1_2 Encodings: TeX, pMML, png Rearrangement (effective with 1.1.7): This equation, which was originally (4.13.3), was moved here, and the symbol for $\omega$ was originally $U$. See also: Annotations for Β§4.13 and Ch.4

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

 4.13.1_3 $T{\mathrm{e}}^{-T}=z.$ β Symbols: $T\left(\NVar{z}\right)$: Tree $T$-function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable Referenced by: Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E1_3 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4

Properties include:

 4.13.2 $\displaystyle W_{0}\left(-{\mathrm{e}}^{-1}\right)$ $\displaystyle=W_{\pm 1}\left(-{\mathrm{e}}^{-1}\mp 0\mathrm{i}\right)=-1,$ $\displaystyle W_{0}\left(0\right)$ $\displaystyle=0,$ $\displaystyle W_{0}\left(\mathrm{e}\right)$ $\displaystyle=1.$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm and $\mathrm{i}$: imaginary unit Permalink: http://dlmf.nist.gov/4.13.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for Β§4.13 and Ch.4
 4.13.3 Moved to (4.13.1_2). β Referenced by: (4.13.1_2) Permalink: http://dlmf.nist.gov/4.13.E3 See also: Annotations for Β§4.13 and Ch.4
 4.13.3_1 $W_{0}\left(x{\mathrm{e}}^{x}\right)=\begin{cases}x,&-1\leq x,\\ \text{(no simpler form)},&x<-1.\end{cases}$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm and $x$: real variable Proof sketch: Take $x=W_{0}\left(t\right)>-1$, with $t>-{\mathrm{e}}^{-1}$. Then $W_{0}\left(x{\mathrm{e}}^{x}\right)=W_{0}\left(W_{0}\left(t\right){\mathrm{e}}% ^{W_{0}\left(t\right)}\right)=W_{0}\left(t\right)=x$. Referenced by: Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E3_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4
 4.13.3_2 $W_{\pm 1}\left(x{\mathrm{e}}^{x}\mp 0\mathrm{i}\right)=\begin{cases}\text{(no % simpler form)},&-1\leq x,\\ x,&x<-1.\end{cases}$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $x$: real variable Proof sketch: Take $x=W_{1}\left(t-0\mathrm{i}\right)<-1$, with $-{\mathrm{e}}^{-1}. Then $W_{1}\left(x{\mathrm{e}}^{x}-0\mathrm{i}\right)=W_{1}\left(W_{1}\left(t-0% \mathrm{i}\right){\mathrm{e}}^{W_{1}\left(t-0\mathrm{i}\right)}-0\mathrm{i}% \right)=W_{1}\left(t-0\mathrm{i}\right)=x$. Referenced by: Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E3_2 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4
 4.13.4 $\frac{\mathrm{d}W}{\mathrm{d}z}=\frac{{\mathrm{e}}^{-W}}{1+W}=\frac{W}{z(1+W)}.$ β
 4.13.4_1 $\frac{{\mathrm{d}}^{n}W}{{\mathrm{d}z}^{n}}=\frac{{\mathrm{e}}^{-nW}p_{n-1}(W)% }{\left(1+W\right)^{2n-1}},$ $n=1,2,3,\dots$, β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\mathrm{e}$: base of natural logarithm, $n$: integer and $z$: complex variable Notes: See Corless et al. (1997, formula (7)). Referenced by: Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E4_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4

in which the $p_{n}(x)$ are polynomials of degree $n$ with

 4.13.4_2 $\displaystyle p_{0}(x)$ $\displaystyle=1,$ $\displaystyle p_{n}(x)$ $\displaystyle=(1+x)p_{n-1}^{\prime}(x)+(1-n(x+3))p_{n-1}(x),$ $n=1,2,3,\dots$. β Symbols: $n$: integer and $x$: real variable Notes: See Corless et al. (1997, formula (8)). Permalink: http://dlmf.nist.gov/4.13.E4_2 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4

Explicit representations for the $p_{n}(x)$ are given in Kalugin and Jeffrey (2011).

 4.13.5 $W_{0}\left(z\right)=\sum_{n=1}^{\infty}\frac{\left(-n\right)^{n-1}}{n!}z^{n},$ $|z|<{\mathrm{e}}^{-1}$.
 4.13.5_1 $\left(\frac{W_{0}\left(z\right)}{z}\right)^{a}={\mathrm{e}}^{-aW_{0}\left(z% \right)}=\sum_{n=0}^{\infty}\frac{a\left(n+a\right)^{n-1}}{n!}\left(-z\right)^% {n},$ $|z|<{\mathrm{e}}^{-1}$, $a\in\mathbb{C}$. β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\mathbb{C}$: complex plane, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $a$: real or complex constant and $z$: complex variable Notes: See Corless et al. (1996, formula (2.36)). Referenced by: Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E5_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4
 4.13.5_2 $\frac{1}{1+W_{0}\left(-z\right)}=\sum_{n=0}^{\infty}\frac{n^{n}}{n!}z^{n},$ $|z|<{\mathrm{e}}^{-1}$. β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable Notes: See Corless et al. (1997, formula (14)). Referenced by: Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E5_2 Encodings: TeX, pMML, png See also: Annotations for Β§4.13 and Ch.4
 4.13.5_3 $\left(1+W_{0}\left(z\right)\right)^{2}=1-2\sum_{n=1}^{\infty}\frac{n^{n-2}}{n!% }\left(-z\right)^{n},$ $|z|<{\mathrm{e}}^{-1}$. β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable Notes: See https://vixra.org/abs/2308.0054. Referenced by: Β§4.13, Erratum (V1.1.11) for Additions Permalink: http://dlmf.nist.gov/4.13.E5_3 Encodings: TeX, pMML, png Addition (effective with 1.1.11): This equation was added. See also: Annotations for Β§4.13 and Ch.4
 4.13.6 $W\left(-{\mathrm{e}}^{-1-(t^{2}/2)}\right)=\sum_{n=0}^{\infty}(-1)^{n-1}c_{n}t% ^{n},$ $|t|<2\sqrt{\pi}$,

where $t\geq 0$ for $W_{0}$, $t\leq 0$ for $W_{\pm 1}$ on the relevant branch cuts,

 4.13.7 $c_{0}=1,\quad c_{1}=1,\quad c_{2}=\tfrac{1}{3},\quad c_{3}=\tfrac{1}{36},\quad c% _{4}=-\tfrac{1}{270},$ β Symbols: $c_{i}$: coefficients Permalink: http://dlmf.nist.gov/4.13.E7 Encodings: TeX, pMML, png See also: Annotations for Β§4.13 and Ch.4
 4.13.8 $c_{n}=\frac{c_{n-1}}{n+1}-\tfrac{1}{2}\sum_{k=2}^{n-1}c_{k}c_{n+1-k},$ $n=2,3,4,\dots$, β Symbols: $k$: integer, $n$: integer and $c_{i}$: coefficients Permalink: http://dlmf.nist.gov/4.13.E8 Encodings: TeX, pMML, png See also: Annotations for Β§4.13 and Ch.4

and

 4.13.9 $1\cdot 3\cdot 5\cdots(2n+1)c_{2n+1}=g_{n},$ β Symbols: $n$: integer, $c_{i}$: coefficients and $g_{n}$: Unknown defined in GA Permalink: http://dlmf.nist.gov/4.13.E9 Encodings: TeX, pMML, png See also: Annotations for Β§4.13 and Ch.4

where $g_{n}$ is defined in Β§5.11(i). See Jeffrey and Murdoch (2017) for an explicit representation for the $c_{n}$ in terms of associated Stirling numbers.

 4.13.9_1 $W_{0}\left(z\right)=\sum_{n=0}^{\infty}d_{n}\left(\mathrm{e}z+1\right)^{\ifrac% {n}{2}},$ $\left|\mathrm{e}z+1\right|<1$, $\left|\operatorname{ph}\left(z+{\mathrm{e}}^{-1}\right)\right|<\pi$, β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative, $\mathrm{e}$: base of natural logarithm, $\operatorname{ph}$: phase, $n$: integer and $z$: complex variable Proof sketch: Substitute (4.13.9_1) into $z(1+W)\frac{\mathrm{d}W}{\mathrm{d}z}=W$. Referenced by: (4.13.9_1), Β§4.13, Β§4.13, Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E9_1 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4

where

 4.13.9_2 $\displaystyle d_{0}$ $\displaystyle=-1,\quad d_{1}=\sqrt{2},\quad d_{2}=-\tfrac{2}{3},\quad d_{3}=% \tfrac{11}{36}\sqrt{2},\quad d_{4}=-\tfrac{43}{135},$ $\displaystyle(n+2)d_{1}d_{n+1}$ $\displaystyle=-2d_{n}+\frac{n}{2}\sum_{k=1}^{n-1}d_{k}d_{n-k}-\frac{n+2}{2}% \sum_{k=1}^{n-1}d_{k+1}d_{n-k+1},$ $n=1,2,3,\dots$. β Symbols: $k$: integer and $n$: integer Permalink: http://dlmf.nist.gov/4.13.E9_2 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4

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

 4.13.10 $W_{k}\left(z\right)\sim\xi_{k}-\ln\xi_{k}+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{% \xi_{k}^{n}}\sum_{m=1}^{n}\genfrac{[}{]}{0.0pt}{}{n}{n-m+1}\frac{\left(-\ln\xi% _{k}\right)^{m}}{m!},$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{k}}$: Stirling cycle number of the first kind, $\sim$: PoincarΓ© asymptotic expansion, $!$: factorial (as in $n!$), $\ln\NVar{z}$: principal branch of logarithm function, $k$: integer, $m$: integer, $n$: integer and $z$: complex variable Notes: See Corless et al. (1996, p. 350) and Jeffrey et al. (1995, Theorem 2). Referenced by: Β§4.13, Β§4.13, Β§4.45(iii), Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E10 Encodings: TeX, pMML, png Addition (effective with 1.1.7): Originally we gave only the first three terms of the infinite series. See also: Annotations for Β§4.13 and Ch.4

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 $z\geq\mathrm{e}$. As $x\to 0-$

 4.13.11 $W_{\pm 1}\left(x\mp 0\mathrm{i}\right)\sim-\eta-\ln\eta+\sum_{n=1}^{\infty}% \frac{1}{\eta^{n}}\sum_{m=1}^{n}\genfrac{[}{]}{0.0pt}{}{n}{n-m+1}\frac{\left(-% \ln\eta\right)^{m}}{m!},$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\genfrac{[}{]}{0.0pt}{}{\NVar{n}}{\NVar{k}}$: Stirling cycle number of the first kind, $\sim$: PoincarΓ© asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ln\NVar{z}$: principal branch of logarithm function, $m$: integer, $n$: integer, $x$: real variable and $\eta$ Notes: See Corless et al. (1996, p. 350). Referenced by: Β§4.13, Erratum (V1.1.2) for Equation (4.13.11), Erratum (V1.1.7) for Expansion Permalink: http://dlmf.nist.gov/4.13.E11 Encodings: TeX, pMML, png Addition (effective with 1.1.7): Originally we gave only the first three terms of the infinite series. See also: Annotations for Β§4.13 and Ch.4

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 $\,\mathrm{d}z=(w+1){\mathrm{e}}^{w}\,\mathrm{d}w$. Examples are

 4.13.12 $\int W\left(z\right)\,\mathrm{d}z=\frac{z}{W\left(z\right)}+zW\left(z\right)-z,$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $z$: complex variable Permalink: http://dlmf.nist.gov/4.13.E12 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4
 4.13.13 $\int\frac{W\left(z\right)}{z}\,\mathrm{d}z=\tfrac{1}{2}W\left(z\right)^{2}+W% \left(z\right),$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral and $z$: complex variable Permalink: http://dlmf.nist.gov/4.13.E13 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4
 4.13.14 $2\int\sin\left(W\left(z\right)\right)\,\mathrm{d}z=z\left(1+\frac{1}{W\left(z% \right)}\right)\sin\left(W\left(z\right)\right)-z\cos\left(W\left(z\right)% \right).$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\cos\NVar{z}$: cosine function, $\,\mathrm{d}\NVar{x}$: differential, $\int$: integral, $\sin\NVar{z}$: sine function and $z$: complex variable Permalink: http://dlmf.nist.gov/4.13.E14 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4
 4.13.15 $W_{0}\left(z\right)=\frac{z}{\pi}\int_{0}^{\pi}\frac{\left(1-t\cot t\right)^{2% }+t^{2}}{z+t{\mathrm{e}}^{-t\cot t}\csc t}\,\mathrm{d}t.$
 4.13.16 $W_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\ln\left(1+z\frac{\sin t}{t}{% \mathrm{e}}^{t\cot t}\right)\,\mathrm{d}t.$ β Symbols: $W\left(\NVar{z}\right)$: Lambert $W$-function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$: cotangent function, $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $\sin\NVar{z}$: sine function and $z$: complex variable Notes: See MezΕ (2020, formula (3)). Permalink: http://dlmf.nist.gov/4.13.E16 Encodings: TeX, pMML, png Addition (effective with 1.1.7): This equation was added. See also: Annotations for Β§4.13 and Ch.4

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

For the foregoing results and further information see Borwein and Corless (1999), Corless et al. (1996), de Bruijn (1961, pp.Β 25β28), Olver (1997b, pp.Β 12β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).