# §4.13 Lambert $W$-Function

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

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

On the $x$-interval $[0,\infty)$ there is one real solution, and it is nonnegative and increasing. On the $x$-interval $(-1/e,0)$ there are two real solutions, one increasing and the other decreasing. We call the solution for which $W\left(x\right)\geq W\left(-1/e\right)$ the principal branch and denote it by $\mathrm{Wp}\left(x\right)$. The other solution is denoted by $\mathrm{Wm}\left(x\right)$. See Figure 4.13.1. Figure 4.13.1: Branches Wp⁡(x) and Wm⁡(x) of the Lambert W-function. A and B denote the points -1/e and e, respectively, on the x-axis. Magnify

Properties include:

 4.13.2 $\displaystyle\mathrm{Wp}\left(-1/e\right)$ $\displaystyle=\mathrm{Wm}\left(-1/e\right)=-1,$ $\displaystyle\mathrm{Wp}\left(0\right)$ $\displaystyle=0,$ $\displaystyle\mathrm{Wp}\left(e\right)$ $\displaystyle=1.$
 4.13.3 $\displaystyle U+\ln U$ $\displaystyle=x,$ $\displaystyle U$ $\displaystyle=U(x)=W\left(e^{x}\right).$
 4.13.4 $\frac{\mathrm{d}W}{\mathrm{d}x}=\frac{e^{-W}}{1+W},$ $x\neq-\dfrac{1}{e}$. ⓘ
 4.13.5 $\mathrm{Wp}\left(x\right)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{n^{n-2}}{(n-1)!}x% ^{n},$ $|x|<\dfrac{1}{e}$.
 4.13.6 $W\left(-e^{-1-(t^{2}/2)}\right)=\sum_{n=0}^{\infty}(-1)^{n-1}c_{n}t^{n},$ $|t|<2\sqrt{\pi}$,

where $t\geq 0$ for $\mathrm{Wp}$, $t\leq 0$ for $\mathrm{Wm}$,

 4.13.7 $c_{0}=1,c_{1}=1,c_{2}=\tfrac{1}{3},c_{3}=\tfrac{1}{36},c_{4}=-\tfrac{1}{270},$ ⓘ Symbols: $c_{i}$: coefficients Permalink: http://dlmf.nist.gov/4.13.E7 Encodings: TeX, pMML, png See also: Annotations for §4.13 and Ch.4
 4.13.8 $c_{n}=\frac{1}{n+1}\left(c_{n-1}-\sum_{k=2}^{n-1}kc_{k}c_{n+1-k}\right),$ $n\geq 2$, ⓘ 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 §4.13 and Ch.4

and

 4.13.9 $1\cdot 3\cdot 5\cdots(2n+1)c_{2n+1}=g_{n},$ ⓘ 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 §4.13 and Ch.4

where $g_{n}$ is defined in §5.11(i).

As $x\to+\infty$

 4.13.10 $\mathrm{Wp}\left(x\right)=\xi-\ln\xi+\frac{\ln\xi}{\xi}+\frac{(\ln\xi)^{2}}{2% \xi^{2}}-\frac{\ln\xi}{\xi^{2}}+O\left(\frac{(\ln\xi)^{3}}{\xi^{3}}\right),$

where $\xi=\ln x$. As $x\to 0-$

 4.13.11 $\mathrm{Wm}\left(x\right)=-\eta-\ln\eta-\frac{\ln\eta}{\eta}+\frac{(\ln\eta)^{% 2}}{2\eta^{2}}-\frac{\ln\eta}{\eta^{2}}+O\left(\frac{(\ln\eta)^{3}}{\eta^{3}}% \right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\mathrm{Wm}\left(\NVar{x}\right)$: nonprincipal branch of Lambert $W$-function, $\ln\NVar{z}$: principal branch of logarithm function, $x$: real variable and $\eta$ Referenced by: Erratum (V1.1.2) for Equation (4.13.11) Permalink: http://dlmf.nist.gov/4.13.E11 Encodings: TeX, pMML, png Correction (effective with 1.1.2): The sign in front of $\frac{(\ln\eta)^{2}}{2\eta^{2}}$, which originally was a minus has been corrected to be a plus. See also: Annotations for §4.13 and Ch.4

where $\eta=\ln\left(-1/x\right)$.

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 integral representations of all branches of the Lambert $W$-function see Kheyfits (2004).

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).