§4.13 Lambert $W$-Function

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

4.13.1		$$W{\mathrm{e}}^W=x.$$
Defines: $W\left(x\right)$: Lambert $W$-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: info for 4.13

On the $x$-interval $[0,\mathrm{\infty })$ there is one real solution, and it is nonnegative and increasing. On the $x$-interval $(-1/\mathrm{e},0)$ there are two real solutions, one increasing and the other decreasing. We call the solution for which $W\left(x\right)\ge W\left(-1/\mathrm{e}\right)$ 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.

Properties include:

4.13.2		$\mathrm{Wp}\left(-1/\mathrm{e}\right)$	$=\mathrm{Wm}\left(-1/\mathrm{e}\right)=-1,$
		$\mathrm{Wp}\left(0\right)$	$=0,$
		$\mathrm{Wp}\left(\mathrm{e}\right)$	$=1.$
Symbols: $\mathrm{Wm}\left(x\right)$: nonprincipal branch of Lambert $W$-function, $\mathrm{Wp}\left(x\right)$: principal branch of Lambert $W$-function and $\mathrm{e}$: base of exponential function Permalink: http://dlmf.nist.gov/4.13.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: info for 4.13

4.13.3		$U+\ln U$	$=x,$
		$U$	$=U(x)=W\left({\mathrm{e}}^x\right).$
Symbols: $W\left(x\right)$: Lambert $W$-function, $\mathrm{e}$: base of exponential function, $\ln z$: principal branch of logarithm function and $x$: real variable Permalink: http://dlmf.nist.gov/4.13.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: info for 4.13

4.13.4		$$\frac{dW}{dx}=\frac{{\mathrm{e}}^{-W}}{1+W},$$
	$x\ne -{\displaystyle \frac{1}{\mathrm{e}}}$.
Symbols: $W\left(x\right)$: Lambert $W$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\mathrm{e}$: base of exponential function and $x$: real variable Referenced by: §4.45(iii) Permalink: http://dlmf.nist.gov/4.13.E4 Encodings: TeX, pMML, png See also: info for 4.13

4.13.5		$$\mathrm{Wp}\left(x\right)=\sum _{n=1}^{\mathrm{\infty }}{(-1)}^{n-1}\frac{n^{n-2}}{(n-1)!}{x}^n,$$
	.
Symbols: $\mathrm{Wp}\left(x\right)$: principal branch of Lambert $W$-function, $\mathrm{e}$: base of exponential function, $!$: factorial (as in $n!$), $n$: integer and $x$: real variable Permalink: http://dlmf.nist.gov/4.13.E5 Encodings: TeX, pMML, png See also: info for 4.13

4.13.6		$$W\left(-{\mathrm{e}}^{-1-(t^2/2)}\right)=\sum _{n=0}^{\mathrm{\infty }}{(-1)}^{n-1}{c}_n{t}^n,$$
	,
Symbols: $W\left(x\right)$: Lambert $W$-function, $\mathrm{e}$: base of exponential function, $n$: integer and $c_{\mathrm{i}}$: coefficients Referenced by: §4.13, §4.45(iii) Permalink: http://dlmf.nist.gov/4.13.E6 Encodings: TeX, pMML, png See also: info for 4.13

where $t\ge 0$ for $\mathrm{Wp}$, $t\le 0$ for $\mathrm{Wm}$,

4.13.7		$$c_0=1,c_1=1,c_2=\frac{1}{3},c_3=\frac{1}{36},c_4=-\frac{1}{270},$$
Symbols: $c_{\mathrm{i}}$: coefficients Permalink: http://dlmf.nist.gov/4.13.E7 Encodings: TeX, pMML, png See also: info for 4.13

4.13.8		$$c_n=\frac{1}{n+1}\left(c_{n-1}-\sum _{k=2}^{n-1}k{c}_k{c}_{n+1-k}\right),$$
	$n\ge 2$,
Symbols: $k$: integer, $n$: integer and $c_{\mathrm{i}}$: coefficients Permalink: http://dlmf.nist.gov/4.13.E8 Encodings: TeX, pMML, png See also: info for 4.13

and

4.13.9		$$1\cdot 3\cdot 5\mathrm{\cdots }(2n+1)c_{2n+1}=g_n,$$
Symbols: $n$: integer, $c_{\mathrm{i}}$: coefficients and $g_n$: Unknown defined in GA Permalink: http://dlmf.nist.gov/4.13.E9 Encodings: TeX, pMML, png See also: info for 4.13

where $g_n$ is defined in §5.11(i).

As $x\to +\mathrm{\infty }$

4.13.10		$$\mathrm{Wp}\left(x\right)=\xi -\ln \xi +\frac{\ln \xi }{\xi }+\frac{{(\ln \xi )}^2}{2{\xi }^2}-\frac{\ln \xi }{{\xi }^2}+O\left(\frac{{(\ln \xi )}^3}{{\xi }^3}\right),$$
Symbols: $O\left(x\right)$: order not exceeding, $\mathrm{Wp}\left(x\right)$: principal branch of Lambert $W$-function, $\ln z$: principal branch of logarithm function and $x$: real variable Referenced by: §4.45(iii) Permalink: http://dlmf.nist.gov/4.13.E10 Encodings: TeX, pMML, png See also: info for 4.13

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: $O\left(x\right)$: order not exceeding, $\mathrm{Wm}\left(x\right)$: nonprincipal branch of Lambert $W$-function, $\ln z$: principal branch of logarithm function, $x$: real variable and $\eta$ Permalink: http://dlmf.nist.gov/4.13.E11 Encodings: TeX, pMML, png See also: info for 4.13

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).