# §4.13 Lambert $\mathop{W\/}\nolimits$-Function

The Lambert $\mathop{W\/}\nolimits$-function $\mathop{W\/}\nolimits\!\left(x\right)$ is the solution of the equation

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

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 $\mathop{W\/}\nolimits\!\left(x\right)\geq\mathop{W\/}\nolimits\!\left(-1/e\right)$ the principal branch and denote it by $\mathop{\mathrm{Wp}\/}\nolimits\!\left(x\right)$. The other solution is denoted by $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)$. See Figure 4.13.1.

Properties include:

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

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

 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
 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

 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

where $g_{n}$ is defined in §5.11(i).

As $x\to+\infty$

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

where $\xi=\mathop{\ln\/}\nolimits x$. As $x\to 0-$

 4.13.11 $\mathop{\mathrm{Wm}\/}\nolimits\!\left(x\right)=-\eta-\mathop{\ln\/}\nolimits% \eta-\frac{\mathop{\ln\/}\nolimits\eta}{\eta}-\frac{(\mathop{\ln\/}\nolimits% \eta)^{2}}{2\eta^{2}}-\frac{\mathop{\ln\/}\nolimits\eta}{\eta^{2}}+\mathop{O\/% }\nolimits\!\left(\frac{(\mathop{\ln\/}\nolimits\eta)^{3}}{\eta^{3}}\right),$

where $\eta=\mathop{\ln\/}\nolimits\!\left(-1/x\right)$.

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 $\mathop{W\/}\nolimits$-function see Kheyfits (2004).

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