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.[3]

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". (Technical note: like the complex logarithm, it is multivalued and thus W is described as the converse relation rather than inverse function.) It is related to the Omega constant, which is equal to W₀(1).

Lambert first considered the related Lambert's Transcendental Equation in 1758,[4] which led to an article by Leonhard Euler in 1783[5] that discussed the special case of we^w.

The equation 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, 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[6]—a fundamental problem in physics. Prompted by this, Rob Corless and developers of the Maple computer algebra system realized that "the Lambert W function has been widely used in many fields, but because of differing notation and the absence of a standard name, awareness of the function was not as high as it should have been."[3][7]

Another example where this function is found is in Michaelis–Menten kinetics.[8]

Although it was widely believed that the Lambert W function cannot be expressed in terms of elementary (Liouvillian) functions, the first published proof did not appear until 2008.[9]

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 ${\displaystyle W(n,ze^{z})=z}$, where z = x + iy. The darker boundaries of a particular region are included in the lighter 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 depends upon the value of z. The value of the integer n changes abruptly when ze^z is at the branch cut of ${\displaystyle W(n,ze^{z})}$, which means that ze^z ≤ 0, except for ${\displaystyle n=0}$ where it is ze^z ≤ −1/e.

Defining ${\displaystyle z=x+iy}$, where x and y are real, and 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 ${\displaystyle W(n,ze^{z})}$ is 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 ${\displaystyle W[n,ze^{z}]}$ is 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 are no discontinuous changes in ${\displaystyle W(n,ze^{z})}$, and those regions 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.[3] Keeping only the first two terms of the expansion,

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

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)).[3]

A number of non-asymptotic bounds are known for the Lambert function.

Hoorfar and Hassani[10] showed that the following bound holds for x ≥ e:

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

They also showed the general bound

${\displaystyle W_{0}(x)\leq \ln \left({\frac {x+y}{1+\ln(y)}}\right),}$

for every ${\displaystyle y>1/e}$ and ${\displaystyle x\geq -1/e}$, with equality only for ${\displaystyle x=y\ln(y)}$. The bound allows many other bounds to be made, such as taking ${\displaystyle y=x+1}$ which gives the bound

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

In 2013 it was proven[11] that the branch W₋₁ can be bounded as follows:

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

Roberto Iacono and John P. Boyd[12] enhanced the bounds as follows:

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

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

The product logarithm Lambert W function W 2(z) plotted in the complex plane from -2-2i to 2+2i

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 real 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:[13]

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

${\displaystyle \ln W_{0}(x)=\ln x-W_{0}(x)\quad {\text{for }}x>0.}$[14]

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

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

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

(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:[15]

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

With Euler's iterated exponential h(x):

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

The following are special values of the principal branch:

${\displaystyle W_{0}\left(-{\frac {\pi }{2}}\right)={\frac {i\pi }{2}}.}$

${\displaystyle W_{0}\left(-{\frac {1}{e}}\right)=-1.}$

${\displaystyle W_{0}\left(2\ln 2\right)=\ln 2.}$

${\displaystyle W_{0}\left(x\ln x\right)=\ln x\ {\text{ provided }}\ x\geqslant 1/e\approx 0.36788.}$

${\displaystyle W_{0}\left(x^{x+1}\ln x\right)=x\ln x\ {\text{ provided }}\ x>0.}$

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

${\displaystyle W_{0}(1)=\Omega =\left(\int _{-\infty }^{\infty }{\frac {dt}{\left(e^{t}-t\right)^{2}+\pi ^{2}}}\right)^{-1}-1\approx 0.56714329\ldots }$ (the omega constant).

${\displaystyle W_{0}(1)=e^{-W_{0}(1)}=\ln \left({\frac {1}{W_{0}(1)}}\right)=-\ln W_{0}(1).}$

${\displaystyle W_{0}(e)=1.}$

${\displaystyle W_{0}\left(e^{1+e}\right)=e.}$

${\displaystyle W_{0}(-1)\approx -0.31813+1.33723i.}$

The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:[16]

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

On the wider domain −1/e ≤ x ≤ e, the considerably simpler representation was found by Mező:[17]

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

Another representation of the principal branch was found by the same author[18] and previously by Kalugin-Jeffrey-Corless:[19]

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

The following continued fraction representation also holds for the principal branch:[20]

${\displaystyle W_{0}(x)={\cfrac {x}{1+{\cfrac {x}{1+{\cfrac {x}{2+{\cfrac {5x}{3+{\cfrac {17x}{10+{\cfrac {133x}{17+{\cfrac {1927x}{190+{\cfrac {13582711x}{94423+\ddots }}}}}}}}}}}}}}}}.}$

Also, if |W₀ (x)| < 1:[21]

${\displaystyle W_{0}(x)={\cfrac {x}{\exp {\cfrac {x}{\exp {\cfrac {x}{\ddots }}}}}}.}$

In turn, if |W₀ (x)| > e, then

${\displaystyle W_{0}(x)=\ln {\cfrac {x}{\ln {\cfrac {x}{\ln {\cfrac {x}{\ddots }}}}}}.}$