By substituting the above equation in $z'=ze^{z))$, we get the defining equation for the W function (and for the W relation in general):
$z'=W(z')e^{W(z')))$
for any complex number z'.
Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W, the complex variable z is then replaced by the real variable x, and the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0). The additional constraint W ≥ −1 defines a single-valued function W_{0}(x). We have W_{0}(0) = 0 and W_{0}(−1/e) = −1. Meanwhile, the lower branch has W ≤ −1 and is denoted W_{−1}(x). It decreases from W_{−1}(−1/e) = −1 to W_{−1}(0^{−}) = −∞.
It can be extended to the function $z=xa^{x))$ using the identity $x={\frac {W[\ln(a)z]}{\ln(a)))$.
The notation convention chosen here (with W_{0} and W_{−1}) follows the canonical reference on the Lambert-W function by Corless, Gonnet, Hare, Jeffrey and Knuth.^{[2]}
History
Lambert first considered the related Lambert's Transcendental Equation in 1758,^{[3]} which led to a paper by Leonhard Euler in 1783^{[4]} that discussed the special case of we^{w}.
The function Lambert considered was
$x=x^{m}+q$
Euler transformed this equation into the form
$x^{a}-x^{b}=(a-b)cx^{a+b))$
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
$\log(x)=cx^{a))$
He then put a = 1 and obtained a convergent series solution.
Taking derivatives with respect to x and after some manipulation the standard form of the Lambert function is obtained.
The Lambert W function was "re-discovered" every decade or so in specialized applications.^{[citation needed]} 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]}
where $L_{1}=\ln(x)$ , $L_{2}=\ln {\bigl (}\ln(x){\bigr )))$ and $\left[{\begin{matrix}\ell +m\\\ell +1\end{matrix))\right]$ is a non-negative Stirling number of the first kind.^{[6]} Keeping only the first two terms of the expansion,
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
$L_{1}=\ln(-x)$ and $L_{2}=\ln(-\ln(-x))$.
In^{[7]} it is shown that the following bound holds for $x\geq e$:
which holds for any $r\in \mathbb {C}$ and $|x|<e^{-1))$.
Identities
A few identities follow from definition:
$W(x\cdot e^{x})=x{\text{ for ))x\geq 0$
$W_{0}(x\cdot e^{x})=x{\text{ for ))x\geq -1$
$W_{-1}(x\cdot e^{x})=x{\text{ for ))x\leq -1$
Note that, since f(x) = x⋅e^{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 x⋅e^{x} = y⋅e^{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), W_{i}(x⋅e^{x}) is the other solution of the equation x⋅e^{x} = y⋅e^{y}.
${\begin{aligned}&W(x)=\ln \left({\frac {x}{W(x)))\right){\text{ for ))x\geq {\frac {-1}{e))\\[5pt]&W\left({\frac {nx^{n)){W(x)^{n-1))}\right)=n\cdot W(x){\text{ for ))n>0{\text{, ))x>0\end{aligned))$
(which can be extended to other n and x if the right branch is chosen)
$W(x)+W(y)=W\left(xy\left({\frac {1}{W(x)))+{\frac {1}{W(y)))\right)\right){\text{ for ))x,y>0$
From inverting f(ln(x)):
$W(x\cdot \ln x)=\ln x{\text{ for ))x>0$
$W(x\cdot \ln x)=W(x)+\ln W(x){\text{ for ))x>0$
With Euler's iterated exponential h(x):
${\begin{aligned}h(x)&=e^{-W(-\ln(x))}\\&={\frac {W(-\ln(x))}{-\ln(x))){\text{ for ))x\not =1\end{aligned))$
Special values
For any non-zero algebraic numberx, W(x) is a transcendental number. Indeed, if W(x) is zero then x must be zero as well, and if W(x) is non-zero 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.
There are countably many branches of the W function, denoted by $W_{k}(z)$, for k an integer; $W_{0}(z)$ being the principal branch. The branch point for the principal branch is at $z=-1/e$, with a branch cut that extends to $-\infty$ along the negative real axis. This branch cut separates the principal branch from the two branches $W_{-1))$ and $W_{1))$. In all other branches, there is a branch point at $z=0$, and a branch cut along the whole negative real axis.
The range of the entire function is the complex plane. The image of the real axis coincides with the quadratrix of Hippias, the parametric curve $w=-t\cot t+it$.
Other formulas
Definite integrals
There are several useful definite integral formulas involving the W function, including the following:
The third identity may be derived from the second by making the substitution $u={\frac {1}{x^{2))))$ and the first can also be derived from the third by the substitution $z=\tan(x)/{\sqrt {2))$.
Except for z along the branch cut $(-\infty ,-1/e]$ (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:
Many equations involving exponentials can be solved using the W function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like Y = Xe^{X} at which point the W function provides the value of the variable in X.
has characteristic equation$\lambda =ae^{-\lambda ))$, leading to $\lambda =W_{k}(a)$ and $y(t)=e^{W_{k}(a)t))$, where $k$ is the branch index. If $a\geq e^{-1))$, only $W_{0}(a)$ need be considered.
Applications
Instrument design
The Lambert-W function has been recently (2013) shown to be the optimal solution for the required magnetic field of a Zeeman slower.^{[15]}
Viscous flows
Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in the laboratory experiments can be described by using the Lambert–Euler omega function as follows:
where H(x) is the debris flow height, x is the channel downstream position, L is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.
Neuroimaging
The Lambert-W function was employed in the field of Neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain voxel, to the corresponding Blood Oxygenation Level Dependent (BOLD) signal.^{[16]}
Chemical engineering
The Lambert-W function was employed in the field of Chemical Engineering for modelling the porous electrode film thickness in a glassy carbon based supercapacitor for electrochemical energy storage. The Lambert "W" function turned out to be the exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.^{[17]}^{[18]}
Materials science
The Lambert-W function was employed in the field of epitaxial film growth for the determination of the critical dislocation onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise the elastic energy stored in the films. Prior to application of Lambert "W" for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert "W" turns it in an explicit equation for analytical handling with ease.^{[19]}
Porous media
The Lambert-W function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneus tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the -1 branch applies if the displacement is unstable with the heavier fluid running underneath the ligther fluid.^{[20]}
The centroid of a set of histograms defined with respect to the symmetrized Kullback-Leibler divergence (also called the Jeffreys divergence) is in closed form using the Lambert function.
The Lambert-W function appears in a quantum-mechanical potential (see The Lambert-W step-potential) which affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potential – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as
$V={\frac {V_{0)){1+W(e^{-x/\sigma })))$.
A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to $z=W(e^{-x/\sigma })$.
The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert W function.^{[21]}
Generalizations
The standard Lambert-W function expresses exact solutions to transcendental algebraic equations (in x) of the form:
$e^{-cx}=a_{o}(x-r)$
(1)
where a_{0}, c and r are real constants. The solution is $x=r+{\frac {1}{c))W\!\left({\frac {c\,e^{-cr)){a_{o))}\right)\,$. Generalizations of the Lambert W function^{[22]}^{[23]}^{[24]} include:
An application to general relativity and quantum mechanics (quantum gravity) in lower dimensions, in fact a previously unknown link (unknown prior to^{[25]}) between these two areas, where the right-hand-side of (1) is now a quadratic polynomial in x:
$e^{-cx}=a_{o}(x-r_{1})(x-r_{2})$
(2)
and where r_{1} and r_{2} are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function has a single argument x but the terms like r_{i} and a_{o} 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 r_{1} = r_{2}, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standard W function. Eq. (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.^{[26]} Here the right-hand-side of (1) (or (2)) is now a ratio of infinite order polynomials in x:
where r_{i} and s_{i} are distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Eq. (3) with its specialized cases expressed in (1) and (2) is related to a large class of delay differential equations. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of (3).^{[27]}
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.^{[28]}
Plots
Plots of the Lambert W function on the complex plane
z = Re(W_{0}(x + iy))
z = Im(W_{0}(x + iy))
W'_{0}(x + iy)
Numerical evaluation
The W function may be approximated using Newton's method,
with successive approximations to $w=W(z)$ (so $z=we^{w))$) being
The Lambert-W function is implemented as
LambertW in Maple, lambertw in GP (and glambertW in PARI), lambertw in MATLAB,^{[29]} also lambertw in octave with the 'specfun' package, as lambert_w in Maxima,^{[30]} as ProductLog (with a silent alias LambertW) in Mathematica,^{[31]} as lambertw in Python scipy's special function package,^{[32]} as LambertW in Perl's ntheory module,^{[33]} and as gsl_sf_lambert_W0 and gsl_sf_lambert_Wm1 functions in special functions section of the GNU Scientific Library – GSL. In R, the Lambert-W function is implemented as the lambertW0 and lambertWm1 functions in the 'lamW' package^{[34]}.
A C++ code for all the branches of the complex Lambert W function is available on the homepage of István Mező.^{[35]}
^Lambert JH, "Observationes variae in mathesin puram", Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III, 128–168, 1758 (facsimile)
^Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921. (facsimile)
^ ^{a}^{b}Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J. (1993). "Lambert's W function in Maple". The Maple Technical Newsletter. 9. MapleTech: 12–22. CiteSeerX10.1.1.33.2556.
^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.
^Dubinov, A. E.; Dubinova, I. D.; Saǐkov, S. K. (2006). The Lambert W Function and Its Applications to Mathematical Problems of Physics (in Russian). RFNC-VNIIEF. p. 53.
^B Ohayon., G Ron. (2013). "New approaches in designing a Zeeman Slower". Journal of Instrumentation. 8 (02): P02016. doi:10.1088/1748-0221/8/02/P02016.
^Sotero, Roberto C.; Iturria-Medina, Yasser (2011). "From Blood oxygenation level dependent (BOLD) signals to brain temperature maps". Bull Math Biol. 73 (11): 2731–47. doi:10.1007/s11538-011-9645-5. PMID21409512.
^Braun, Artur; Wokaun, Alexander; Hermanns, Heinz-Guenter (2003). "Analytical Solution to a Growth Problem with Two Moving Boundaries". Appl Math Model. 27 (1): 47–52. doi:10.1016/S0307-904X(02)00085-9.
^Braun, Artur; Baertsch, Martin; Schnyder, Bernhard; Koetz, Ruediger (2000). "A Model for the film growth in samples with two moving boundaries – An Application and Extension of the Unreacted-Core Model". Chem Eng Sci. 55 (22): 5273–5282. doi:10.1016/S0009-2509(00)00143-3.
^Braun, Artur; Briggs, Keith M.; Boeni, Peter (2003). "Analytical solution to Matthews' and Blakeslee's critical dislocation formation thickness of epitaxially grown thin films". J Cryst Growth. 241 (1/2): 231–234. Bibcode:2002JCrGr.241..231B. doi:10.1016/S0022-0248(02)00941-7.
^Colla, Pietro (2014). "A New Analytical Method for the Motion of a Two-Phase Interface in a Tilted Porous Medium". PROCEEDINGS,Thirty-Eighth Workshop on Geothermal Reservoir Engineering,Stanford University. SGP-TR-202.([1])
^de la Madrid, R. (2017). "Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential". Nucl. Phys. A. 962: 24–45. arXiv:1704.00047. doi:10.1016/j.nuclphysa.2017.03.006.
^Scott, T. C.; Mann, R. B.; Martinez Ii, Roberto E. (2006). "General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function". AAECC (Applicable Algebra in Engineering, Communication and Computing). 17 (1): 41–47. arXiv:math-ph/0607011. doi:10.1007/s00200-006-0196-1.
^Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion". Chem. Phys. 324 (2–3): 323–338. arXiv:physics/0607081. doi:10.1016/j.chemphys.2005.10.031.
^Maignan, Aude; Scott, T. C. (2016). "Fleshing out the Generalized Lambert W Function". SIGSAM. 50 (2): 45–60. doi:10.1145/2992274.2992275.
^Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions". Phys. Rev. A. 75 (6): 060101. doi:10.1103/PhysRevA.75.060101.
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.