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4 Elementary FunctionsLogarithm, Exponential, Powers

Β§4.13 Lambert W-Function

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

4.13.1 W⁒eW=z.

On the z-interval [0,∞) there is one real solution, and it is nonnegative and increasing. On the z-interval (βˆ’eβˆ’1,0) there are two real solutions, one increasing and the other decreasing. We call the increasing solution for which W⁑(z)β‰₯W⁑(βˆ’eβˆ’1)=βˆ’1 the principal branch and denote it by W0⁑(z). See Figure 4.13.1.

See accompanying text
Figure 4.13.1: Branches W0⁑(x), WΒ±1⁑(xβˆ“0⁒i) of the Lambert W-function. Magnify

The decreasing solution can be identified as WΒ±1⁑(xβˆ“0⁒i). Other solutions of (4.13.1) are other branches of W⁑(z). They are denoted by Wk⁑(z), kβˆˆβ„€, and have the property

4.13.1_1 Wk⁑(z)=lnk⁒(z)βˆ’ln⁑(lnk⁒(z))+o⁑(1),

where lnk⁒(z)=ln⁑(z)+2⁒π⁒i⁒k. W0⁑(z) is a single-valued analytic function on β„‚βˆ–(βˆ’βˆž,βˆ’eβˆ’1], real-valued when z>βˆ’eβˆ’1, and has a square root branch point at z=βˆ’eβˆ’1. See (4.13.6) and (4.13.9_1). The other branches Wk⁑(z) are single-valued analytic functions on β„‚βˆ–(βˆ’βˆž,0], have a logarithmic branch point at z=0, and, in the case k=Β±1, have a square root branch point at z=βˆ’eβˆ’1βˆ“0⁒i respectively. See Figure 4.13.2.

See accompanying text
Figure 4.13.2: The W⁑(z) function on the first 5 Riemann sheets. W⁑(z) maps the first Riemann sheet |ph⁑(z+eβˆ’1)|<Ο€ in the middle of the left-hand side to the region enclosed by the green curve on the right-hand side; it maps the Riemann sheet Ο€<ph⁑z<3⁒π on the left-hand side to the region enclosed by the pink, green and orange curves on the right-hand side, etc. Magnify

Alternative notations are Wp⁑(x) for W0⁑(x), Wm⁑(x) for Wβˆ’1⁑(x+0⁒i), both previously used in this section, the Wright Ο‰-function ω⁑(z)=W⁑(ez), which is single-valued, satisfies

4.13.1_2 ω⁑(z)+ln⁑(ω⁑(z))=z,

and has several advantages over the Lambert W-function (see Lawrence et al. (2012)), and the tree T-function T⁑(z)=βˆ’W⁑(βˆ’z), which is a solution of

4.13.1_3 T⁒eβˆ’T=z.

Properties include:

4.13.2 W0⁑(βˆ’eβˆ’1) =WΒ±1⁑(βˆ’eβˆ’1βˆ“0⁒i)=βˆ’1,
W0⁑(0) =0,
W0⁑(e) =1.
4.13.3 Moved to (4.13.1_2).
4.13.3_1 W0⁑(x⁒ex)={x,βˆ’1≀x,(no simpler form),x<βˆ’1.
4.13.3_2 WΒ±1⁑(x⁒exβˆ“0⁒i)={(no simpler form),βˆ’1≀x,x,x<βˆ’1.
4.13.4 dWdz=eβˆ’W1+W=Wz⁒(1+W).
4.13.4_1 dnWdzn=eβˆ’n⁒W⁒pnβˆ’1⁒(W)(1+W)2⁒nβˆ’1,

in which the pn⁒(x) are polynomials of degree n with

4.13.4_2 p0⁒(x) =1,
pn⁒(x) =(1+x)⁒pnβˆ’1′⁒(x)+(1βˆ’n⁒(x+3))⁒pnβˆ’1⁒(x),

Explicit representations for the pn⁒(x) are given in Kalugin and Jeffrey (2011).

4.13.5 W0⁑(z)=βˆ‘n=1∞(βˆ’n)nβˆ’1n!⁒zn,
4.13.5_1 (W0⁑(z)z)a=eβˆ’a⁒W0⁑(z)=βˆ‘n=0∞a⁒(n+a)nβˆ’1n!⁒(βˆ’z)n,
|z|<eβˆ’1, aβˆˆβ„‚.
4.13.5_2 11+W0⁑(βˆ’z)=βˆ‘n=0∞nnn!⁒zn,
4.13.5_3 (1+W0⁑(z))2=1βˆ’2β’βˆ‘n=1∞nnβˆ’2n!⁒(βˆ’z)n,
4.13.6 W⁑(βˆ’eβˆ’1βˆ’(t2/2))=βˆ‘n=0∞(βˆ’1)nβˆ’1⁒cn⁒tn,

where tβ‰₯0 for W0, t≀0 for WΒ±1 on the relevant branch cuts,

4.13.7 c0=1,c1=1,c2=13,c3=136,c4=βˆ’1270,
4.13.8 cn=cnβˆ’1n+1βˆ’12β’βˆ‘k=2nβˆ’1ck⁒cn+1βˆ’k,


4.13.9 1β‹…3β‹…5⁒⋯⁒(2⁒n+1)⁒c2⁒n+1=gn,

where gn is defined in Β§5.11(i). See Jeffrey and Murdoch (2017) for an explicit representation for the cn in terms of associated Stirling numbers.

4.13.9_1 W0⁑(z)=βˆ‘n=0∞dn⁒(e⁒z+1)n/2,
|e⁒z+1|<1, |ph⁑(z+eβˆ’1)|<Ο€,


4.13.9_2 d0 =βˆ’1,d1=2,d2=βˆ’23,d3=1136⁒2,d4=βˆ’43135,
(n+2)⁒d1⁒dn+1 =βˆ’2⁒dn+n2β’βˆ‘k=1nβˆ’1dk⁒dnβˆ’kβˆ’n+22β’βˆ‘k=1nβˆ’1dk+1⁒dnβˆ’k+1,

For the definition of Stirling cycle numbers of the first kind [nk] see (26.13.3). As |z|β†’βˆž

4.13.10 Wk⁑(z)∼ξkβˆ’ln⁑ξk+βˆ‘n=1∞(βˆ’1)nΞΎknβ’βˆ‘m=1n[nnβˆ’m+1]⁒(βˆ’ln⁑ξk)mm!,

where ΞΎk=ln⁑(z)+2⁒π⁒i⁒k. For large enough |z| 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 zβ‰₯e. As xβ†’0βˆ’

4.13.11 WΒ±1⁑(xβˆ“0⁒i)βˆΌβˆ’Ξ·βˆ’ln⁑η+βˆ‘n=1∞1Ξ·nβ’βˆ‘m=1n[nnβˆ’m+1]⁒(βˆ’ln⁑η)mm!,

where Ξ·=ln⁑(βˆ’1/x). For these results and other asymptotic expansions see Corless et al. (1997).

For integrals of W⁑(z) use the substitution w=W⁑(z), z=w⁒ew and dz=(w+1)⁒ew⁒dw. Examples are

4.13.12 ∫W⁑(z)⁒dz=zW⁑(z)+z⁒W⁑(z)βˆ’z,
4.13.13 ∫W⁑(z)z⁒dz=12⁒W⁑(z)2+W⁑(z),
4.13.14 2⁒∫sin⁑(W⁑(z))⁒dz=z⁒(1+1W⁑(z))⁒sin⁑(W⁑(z))βˆ’z⁒cos⁑(W⁑(z)).
4.13.15 W0⁑(z)=zΟ€β’βˆ«0Ο€(1βˆ’t⁒cot⁑t)2+t2z+t⁒eβˆ’t⁒cot⁑t⁒csc⁑t⁒dt.
4.13.16 W0⁑(z)=1Ο€β’βˆ«0Ο€ln⁑(1+z⁒sin⁑tt⁒et⁒cot⁑t)⁒dt.

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 25–28), Olver (1997b, pp.Β 12–13), 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).