Lambert W function

The Lambert W function is named after Johann Heinrich Lambert. The principal branch W₀ is denoted Wp in the Digital Library of Mathematical Functions, and the branch W₋₁ is denoted Wm there.

The notation convention chosen here (with W₀ and W₋₁) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth.[2]

The name "product logarithm" can be understood as this: Since the inverse function of f(w) = e^w is called the logarithm, it makes sense to call the inverse function of the product we^w as "product logarithm". It is related to the Omega constant, which is equal to W₀(1).

Lambert first considered the related Lambert's Transcendental Equation in 1758,[3] which led to an article by Leonhard Euler in 1783[4] that discussed the special case of we^w.

The function Lambert considered was

${\displaystyle x=x^{m}+q.}$

Euler transformed this equation into the form

${\displaystyle x^{a}-x^{b}=(a-b)cx^{a+b}.}$

Both authors derived a series solution for their equations.

Once Euler had solved this equation he considered the case a = b. Taking limits he derived the equation

${\displaystyle \ln x=cx^{a}.}$

He then put a = 1 and obtained a convergent series solution for the resulting equation, expressing x in terms of c.

After taking derivatives with respect to x and some manipulation, the standard form of the Lambert function is obtained.

In 1993, when it was reported that the Lambert W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem in physics—Corless and developers of the Maple computer algebra system made a library search and found that this function was ubiquitous in nature.[2][5]

Another example where this function is found is in Michaelis–Menten kinetics.

Although it was folklore knowledge that the Lambert W function cannot be expressed in terms of elementary (Liouvillian) functions, the first published proof did not appear until 2008.[6]

The range of the W function, showing all branches. The black curves (including the real axis) form the image of the real axis, the orange curves are the image of the imaginary axis. The purple curve is the image of a small circle around the point z = 0; the red curves are the image of a small circle around the point z = −1/e.

Plot of the imaginary part of W[n,x+i y] for branches n=-2,-1,0,1,2. The plot is similar to that of the multivalued complex logarithm function except that the spacing between sheets is not constant and the connection of the principal sheet is different

There are countably many branches of the W function, denoted by W_k(z), for integer k; W₀(z) being the main (or principal) branch. W₀(z) is defined for all complex numbers z while W_k(z) with k ≠ 0 is defined for all non-zero z. We have W₀(0) = 0 and limz→0 W_k(z) = −∞ for all k ≠ 0.

The branch point for the principal branch is at z = −1/e, with a branch cut that extends to −∞ along the negative real axis. This branch cut separates the principal branch from the two branches W₋₁ and W₁. In all branches W_k with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis.

The functions W_k(z), k ∈ Z are all injective and their ranges are disjoint. The range of the entire multivalued function W is the complex plane. The image of the real axis is the union of the real axis and the quadratrix of Hippias, the parametric curve w = −t cot t + it.

Inverse

Regions of the complex plane for which ${\displaystyle W(n,ze^{z})=z}$. The darker boundaries of a particular region are included in the lighter colored region of the same color. The point at {-1,0} is included in both the ${\displaystyle n=-1}$ (blue) region and the ${\displaystyle n=0}$ (gray) region. Horizontal grid lines are in multiples of π.

The range plot above also delineates the regions in the complex plane where the simple inverse relationship '${\displaystyle W(n,ze^{z})=z}$ is true. f=ze^z implies that there exists an n such that ${\displaystyle z=W(n,f)=W(n,ze^{z})}$, where n will depend upon the value of z. The value of the integer n will change abruptly when ze^z is at the branch cut of ${\displaystyle W(n,ze^{z})}$ which will mean that ze^z ≤ 0, except for ${\displaystyle n=0}$ where it will be ze^z ≤ -1/e.

Define ${\displaystyle z=x+iy}$ where x and y are real. Expressing e^z in polar coordinates, it is seen that:

${\displaystyle {\begin{aligned}ze^{z}&=(x+iy)e^{x}(\cos(y)+i\sin(y))\\&=e^{x}(x\cos(y)-y\sin(y))+ie^{x}(x\sin(y)+y\cos(y))\\\end{aligned}}}$

For ${\displaystyle n\neq 0}$, the branch cut for ${\displaystyle W[n,ze^{z}]}$ will be the non-positive real axis so that:

${\displaystyle x\sin(y)+y\cos(y)=0\Rightarrow x=-y/\tan(y)}$

and

${\displaystyle (x\cos(y)-y\sin(y))e^{x}\leq 0}$

For ${\displaystyle n=0}$, the branch cut for ${\displaystyle W[n,ze^{z}]}$ will be the real axis with ${\displaystyle -\infty <z\leq -1/e}$ so that the inequality becomes:

${\displaystyle (x\cos(y)-y\sin(y))e^{x}\leq -1/e}$

Inside the regions bounded by the above, there will be no discontinuous changes in ${\displaystyle W(n,ze^{z})}$ and those regions will specify where the W function is simply invertible: i.e. ${\displaystyle W(n,ze^{z})=z}$.

The Taylor series of W₀ around 0 can be found using the Lagrange inversion theorem and is given by

${\displaystyle W_{0}(x)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}x^{n}=x-x^{2}+{\tfrac {3}{2}}x^{3}-{\tfrac {8}{3}}x^{4}+{\tfrac {125}{24}}x^{5}-\cdots .}$

The radius of convergence is 1/e, as may be seen by the ratio test. The function defined by this series can be extended to a holomorphic function defined on all complex numbers with a branch cut along the interval (−∞, −1/e]; this holomorphic function defines the principal branch of the Lambert W function.

For large values of x, W₀ is asymptotic to

${\displaystyle {\begin{aligned}W_{0}(x)&=L_{1}-L_{2}+{\frac {L_{2}}{L_{1}}}+{\frac {L_{2}\left(-2+L_{2}\right)}{2L_{1}^{2}}}+{\frac {L_{2}\left(6-9L_{2}+2L_{2}^{2}\right)}{6L_{1}^{3}}}+{\frac {L_{2}\left(-12+36L_{2}-22L_{2}^{2}+3L_{2}^{3}\right)}{12L_{1}^{4}}}+\cdots \\[5pt]&=L_{1}-L_{2}+\sum _{l=0}^{\infty }\sum _{m=1}^{\infty }{\frac {(-1)^{l}\left[{\begin{smallmatrix}l+m\\l+1\end{smallmatrix}}\right]}{m!}}L_{1}^{-l-m}L_{2}^{m},\end{aligned}}}$

where L₁ = ln x, L₂ = ln ln x, and [^{l + m}
_{l + 1}] is a non-negative Stirling number of the first kind.[2] Keeping only the first two terms of the expansion,

${\displaystyle W_{0}(x)=\ln x-\ln \ln x+o(1).}$

The other real branch, W₋₁, defined in the interval [−1/e, 0), has an approximation of the same form as x approaches zero, with in this case L₁ = ln(−x) and L₂ = ln(−ln(−x)).[2]

It is shown[7] that the following bound holds (the upper bound only for x ≥ e):

${\displaystyle \ln x-\ln \ln x+{\frac {\ln \ln x}{2\ln x}}\leq W_{0}(x)\leq \ln x-\ln \ln x+{\frac {e}{e-1}}{\frac {\ln \ln x}{\ln x}}.}$

In 2013 it was proven[8] that the branch W₋₁ can be bounded as follows:

${\displaystyle -1-{\sqrt {2u}}-u<W_{-1}\left(-e^{-u-1}\right)<-1-{\sqrt {2u}}-{\tfrac {2}{3}}u\quad {\text{for }}u>0.}$

A plot of W_j(x e^x) where blue is for j=0 and red is for j=-1. The diagonal line represents the intervals where W_j(x e^x)=x

A few identities follow from the definition:

${\displaystyle {\begin{aligned}W_{0}(xe^{x})&=x&{\text{for }}x&\geq -1,\\W_{-1}(xe^{x})&=x&{\text{for }}x&\leq -1.\end{aligned}}}$

Note that, since f(x) = xe^x is not injective, it does not always hold that W(f(x)) = x, much like with the inverse trigonometric functions. For fixed x < 0 and x ≠ −1, the equation xe^x = ye^y has two solutions in y, one of which is of course y = x. Then, for i = 0 and x < −1, as well as for i = −1 and x ∈ (−1, 0), y = W_i(xe^x) is the other solution.

Some other identities:[9]

${\displaystyle {\begin{aligned}&W(x)\cdot e^{W(x)}=x,\quad {\text{therefore:}}\\[5pt]&e^{W(x)}={\frac {x}{W(x)}},\qquad e^{-W(x)}={\frac {W(x)}{x}},\qquad e^{nW(x)}=\left({\frac {x}{W(x)}}\right)^{n}.\end{aligned}}}$

${\displaystyle \ln W_{0}(x)=\ln x-W_{0}(x)\quad {\text{for }}x>0.}$[10]

${\displaystyle W_{0}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{0}\left(x\ln x\right)}=x\quad {\text{for }}{\frac {1}{e}}\leq x.}$

${\displaystyle W_{-1}\left(x\ln x\right)=\ln x\quad {\text{and}}\quad e^{W_{-1}\left(x\ln x\right)}=x\quad {\text{for }}0<x\leq {\frac {1}{e}}.}$

${\displaystyle {\begin{aligned}&W(x)=\ln {\frac {x}{W(x)}}&&{\text{for }}x\geq -{\frac {1}{e}},\\[5pt]&W\left({\frac {nx^{n}}{W\left(x\right)^{n-1}}}\right)=nW(x)&&{\text{for }}n,x>0\end{aligned}}}$

(which can be extended to other n and x if the correct branch is chosen).

${\displaystyle W(x)+W(y)=W\left(xy\left({\frac {1}{W(x)}}+{\frac {1}{W(y)}}\right)\right)\quad {\text{for }}x,y>0.}$

Substituting −ln x in the definition:

${\displaystyle {\begin{aligned}W_{0}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}0&<x\leq e,\\[5pt]W_{-1}\left(-{\frac {\ln x}{x}}\right)&=-\ln x&{\text{for }}x&>e.\end{aligned}}}$

With Euler's iterated exponential h(x):

${\displaystyle {\begin{aligned}h(x)&=e^{-W(-\ln x)}\\&={\frac {W(-\ln x)}{-\ln x}}\quad {\text{for }}x\neq 1.\end{aligned}}}$

For any nonzero algebraic number x, W(x) is a transcendental number. Indeed, if W(x) is zero, then x must be zero as well, and if W(x) is nonzero and algebraic, then by the Lindemann–Weierstrass theorem, e^W(x) must be transcendental, implying that x = W(x)e^W(x) must also be transcendental.

The following are special values of the principal branch:

${\displaystyle W\left(-{\frac {\pi }{2}}\right)={\frac {i\pi }{2}}.}$

${\displaystyle W\left(-{\frac {1}{e}}\right)=-1.}$

${\displaystyle W\left(-{\frac {\ln a}{a}}\right)=-\ln a\quad \left({\frac {1}{e}}\leq a\leq e\right).}$

${\displaystyle W\left(2\ln 2\right)=\ln 2.}$

${\displaystyle W(0)=0.}$

${\displaystyle W(1)=\Omega =\left(\int _{-\infty }^{\infty }{\frac {dt}{\left(e^{t}-t\right)^{2}+\pi ^{2}}}\right)^{-1}-1\approx 0.56714329\ldots }$ (the omega constant).

${\displaystyle W(1)=e^{-W(1)}=\ln \left({\frac {1}{W(1)}}\right)=-\ln W(1).}$

${\displaystyle W(e)=1.}$

${\displaystyle W\left(e^{1+e}\right)=e.}$

${\displaystyle W(-1)\approx -0.31813+1.33723i.}$

The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:[11]

${\displaystyle -{\frac {\pi }{2}}W(-x)=\int _{0}^{\pi }{\frac {\sin \left({\tfrac {3}{2}}t\right)-xe^{\cos t}\sin \left({\tfrac {5}{2}}t-\sin t\right)}{1-2xe^{\cos t}\cos(t-\sin t)+x^{2}e^{2\cos t}}}\sin \left({\tfrac {1}{2}}t\right)\,dt\quad {\text{for }}|x|<{\frac {1}{e}}.}$

On the wider domain −1/e ≤ x ≤ e, the considerably simpler representation is found by Mező:[12]

${\displaystyle W(x)={\frac {1}{\pi }}\operatorname {Re} \int _{0}^{\pi }\ln \left({\frac {e^{e^{it}}-xe^{-it}}{e^{e^{it}}-xe^{it}}}\right)\,dt.}$

Another representation of the principal branch was found by the same author:[13]

${\displaystyle W(x)={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+x{\frac {\sin t}{t}}e^{t\cot t}\right)dt.}$

The following continued fraction representation also holds for the principal branch:[14]

${\displaystyle W(x)={\cfrac {x}{1+{\cfrac {x}{1+{\cfrac {x}{2+{\cfrac {5x}{3+{\cfrac {17x}{10+{\cfrac {133x}{17+{\cfrac {1927x}{190+{\cfrac {13582711x}{94423+\ddots }}}}}}}}}}}}}}}}.}$

Also, if |W(z)| < 1:[15]

${\displaystyle W(x)={\cfrac {x}{\exp {\cfrac {x}{\exp {\cfrac {x}{\ddots }}}}}}.}$

In turn, if |W(z)| > e, then

${\displaystyle W(x)=\ln {\cfrac {x}{\ln {\cfrac {x}{\ln {\cfrac {x}{\ddots }}}}}}.}$

The standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:

${\displaystyle e^{-cx}=a_{0}(x-r)}$

(1)

where a₀, c and r are real constants. The solution is

${\displaystyle x=r+{\frac {1}{c}}W\left({\frac {c\,e^{-cr}}{a_{0}}}\right).}$

Generalizations of the Lambert W function[32][33][34] include:

• An application to general relativity and quantum mechanics (quantum gravity) in lower dimensions, in fact a link (unknown prior to 2007[35]) between these two areas, where the right-hand side of (1) is replaced by a quadratic polynomial in x:

${\displaystyle e^{-cx}=a_{0}\left(x-r_{1}\right)\left(x-r_{2}\right),}$

(2)

where r₁ and r₂ are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which has a single argument x but the terms like r_i and a₀ are parameters of that function. In this respect, the generalization resembles the hypergeometric function and the Meijer G function but it belongs to a different class of functions. When r₁ = r₂, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standard W function. Equation (2) expresses the equation governing the dilaton field, from which is derived the metric of the R = T or lineal two-body gravity problem in 1 + 1 dimensions (one spatial dimension and one time dimension) for the case of unequal rest masses, as well as the eigenenergies of the quantum-mechanical double-well Dirac delta function model for unequal charges in one dimension.

• Analytical solutions of the eigenenergies of a special case of the quantum mechanical three-body problem, namely the (three-dimensional) hydrogen molecule-ion.[36] Here the right-hand side of (1) is replaced by a ratio of infinite order polynomials in x:

${\displaystyle e^{-cx}=a_{0}{\frac {\displaystyle \prod _{i=1}^{\infty }(x-r_{i})}{\displaystyle \prod _{i=1}^{\infty }(x-s_{i})}}}$

(3)

where r_i and s_i are distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Equation (3) with its specialized cases expressed in (1) and (2) is related to a large class of delay differential equations. G. H. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of (3).[37]

Applications of the Lambert W function in fundamental physical problems are not exhausted even for the standard case expressed in (1) as seen recently in the area of atomic, molecular, and optical physics.[38]

• Plots of the Lambert W function on the complex plane
• 
  z = Re(W₀(x + iy))
• 
  z = Im(W₀(x + iy))
• 
  z = |W₀(x + iy)|
• 
  Superimposition of the previous three plots

The W function may be approximated using Newton's method, with successive approximations to w = W(z) (so z = we^w) being

${\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}+w_{j}e^{w_{j}}}}.}$

The W function may also be approximated using Halley's method,

${\displaystyle w_{j+1}=w_{j}-{\frac {w_{j}e^{w_{j}}-z}{e^{w_{j}}\left(w_{j}+1\right)-{\dfrac {\left(w_{j}+2\right)\left(w_{j}e^{w_{j}}-z\right)}{2w_{j}+2}}}}}$

given in Corless et al.[2] to compute W.

The Lambert W function is implemented as LambertW in Maple, lambertw in GP (and glambertW in PARI), lambertw in Matlab,[39] also lambertw in Octave with the specfun package, as lambert_w in Maxima,[40] as ProductLog (with a silent alias LambertW) in Mathematica,[41] as lambertw in Python scipy's special function package,[42] as LambertW in Perl's ntheory module,[43] and as gsl_sf_lambert_W0, gsl_sf_lambert_Wm1 functions in the special functions section of the GNU Scientific Library (GSL). In the Boost C++ libraries, the calls are lambert_w0, lambert_wm1, lambert_w0_prime, and lambert_wm1_prime. In R, the Lambert W function is implemented as the lambertW0 and lambertWm1 functions in the lamW package.[44]

A C++ code for all the branches of the complex Lambert W function is available on the homepage of István Mező.[45]

• Wright Omega function
• Lambert's trinomial equation
• Lagrange inversion theorem
• Experimental mathematics
• Holstein–Herring method
• R = T model
• Ross' π lemma

• Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.; Knuth, Donald (1996). "On the Lambert W function" (PDF). Advances in Computational Mathematics. 5: 329–359. doi:10.1007/BF02124750. ISSN 1019-7168. S2CID 29028411. Archived from the original (PDF) on 2010-12-14. Retrieved 2007-03-10.
• Chapeau-Blondeau, F.; Monir, A. (2002). "Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2" (PDF). IEEE Trans. Signal Process. 50 (9). doi:10.1109/TSP.2002.801912. Archived from the original (PDF) on 2012-03-28. Retrieved 2004-03-10.
• Francis; et al. (2000). "Quantitative General Theory for Periodic Breathing". Circulation. 102 (18): 2214–21. CiteSeerX 10.1.1.505.7194. doi:10.1161/01.cir.102.18.2214. PMID 11056095. S2CID 14410926. (Lambert function is used to solve delay-differential dynamics in human disease.)
• Hayes, B. (2005). "Why W?" (PDF). American Scientist. 93 (2): 104–108. doi:10.1511/2005.2.104.
• Roy, R.; Olver, F. W. J. (2010), "Lambert W function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• Stewart, Seán M. (2005). "A New Elementary Function for Our Curricula?" (PDF). Australian Senior Mathematics Journal. 19 (2): 8–26. ISSN 0819-4564. ERIC EJ720055. Lay summary.
• Veberic, D., "Having Fun with Lambert W(x) Function" arXiv:1003.1628 (2010); Veberic, D. (2012). "Lambert W function for applications in physics". Computer Physics Communications. 183 (12): 2622–2628. arXiv:1209.0735. Bibcode:2012CoPhC.183.2622V. doi:10.1016/j.cpc.2012.07.008. S2CID 315088.
• Chatzigeorgiou, I. (2013). "Bounds on the Lambert function and their Application to the Outage Analysis of User Cooperation". IEEE Communications Letters. 17 (8): 1505–1508. arXiv:1601.04895. doi:10.1109/LCOMM.2013.070113.130972. S2CID 10062685.

Wikimedia Commons has media related to Lambert W function.

• National Institute of Science and Technology Digital Library – Lambert W
• MathWorld – Lambert W-Function
• Computing the Lambert W function
• Corless et al. Notes about Lambert W research
• GPL C++ implementation with Halley's and Fritsch's iteration.
• Special Functions of the GNU Scientific Library – GSL