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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.13 Lambert W-Function

The Lambert W-function W(x) is the solution of the equation

4.13.1 WeW=x.

On the x-interval [0,) 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(x)W(-1/e) the principal branch and denote it by Wp(x). The other solution is denoted by Wm(x). See Figure 4.13.1.

See accompanying text
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 Wp(-1/e) =Wm(-1/e)
=-1,
Wp(0) =0,
Wp(e) =1.
4.13.3 U+lnU =x,
U =U(x)
=W(ex).
4.13.4 dWdx=e-W1+W,
x-1e.
4.13.5 Wp(x)=n=1(-1)n-1nn-2(n-1)!xn,
|x|<1e.
4.13.6 W(-e-1-(t2/2))=n=0(-1)n-1cntn,
|t|<2π,

where t0 for Wp, t0 for Wm,

4.13.7 c0=1,c1=1,c2=13,c3=136,c4=-1270,
4.13.8 cn=1n+1(cn-1-k=2n-1kckcn+1-k),
n2,

and

4.13.9 135(2n+1)c2n+1=gn,

where gn is defined in §5.11(i).

As x+

4.13.10 Wp(x)=ξ-lnξ+lnξξ+(lnξ)22ξ2-lnξξ2+O((lnξ)3ξ3),

where ξ=lnx. As x0-

4.13.11 Wm(x)=-η-lnη-lnηη-(lnη)22η2-lnηη2+O((lnη)3η3),

where η=ln(-1/x).

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).