Lambert W function
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function f(z) = zez where ez is the exponential function and z is any complex number. In other words
By substituting in the above equation, we get the defining equation for the W function (and for the W relation in general):
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 W0(x). We have W0(0) = 0 and W0(−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−) = −∞.
The Lambert W relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. the maxima of the Planck, Bose–Einstein, and Fermi–Dirac distributions) and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1). In biochemistry, and in particular enzyme kinetics, a closed-form solution for the time course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.
- 1 Terminology
- 2 History
- 3 Calculus
- 4 Asymptotic expansions
- 5 Identities
- 6 Special values
- 7 Other formulas
- 8 Applications
- 9 Generalizations
- 10 Plots
- 11 Numerical evaluation
- 12 Software
- 13 See also
- 14 Notes
- 15 References
- 16 External links
The Lambert W function was "re-discovered" every decade or so in specialized applications. 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.
(W is not differentiable for z = −1/e.) As a consequence, we get the following formula for the derivative of W:
Using the identity , we get the following equivalent formula which holds for all :
(The last equation is more common in the literature but does not hold at .)
One consequence of which (using the fact that ) is the identity:
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
The other real branch, , defined in the interval [−1/e, 0), has an approximation of the same form as x approaches zero, with in this case and .
In  it is shown that the following bound holds for :
In  it was proven that branch can be bounded as follows:
Integer and complex powers
Integer powers of also admit simple Taylor (or Laurent) series expansions at
More generally, for the Lagrange inversion formula gives
which is, in general, a Laurent series of order r. Equivalently, the latter can be written in the form of a Taylor expansion of powers of
which holds for any and .
A few identities follow from definition:
Note that, since f(x) = x⋅ex is not injective, not always W(f(x)) = x. For fixed x < 0 and x ≠ 1 the equation x⋅ex = y⋅ey 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), Wi(x⋅ex) is the other solution of the equation x⋅ex = y⋅ey.
From inverting f(ln(x)):
With Euler's iterated exponential h(x):
For any non-zero 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 non-zero and algebraic, then by the Lindemann–Weierstrass theorem, eW(x) must be transcendental, implying that x=W(x)eW(x) must also be transcendental.
- (the Omega constant)
There are several useful integration formulas involving the W function. Some of these include the following:
The second identity can be derived by making the substitution
The third identity may be derived from the second by making the substitution and the first can be derived from the third by the substitution .
Except for z along the branch cut (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:
where the two integral expressions are equivalent due to the symmetry of the integrand.
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 = XeX at which point the W function provides the value of the variable in X.
In other words :
More generally, the equation
can be transformed via the substitution
which yields the final solution
Whenever the complex infinite exponential tetration
converges, the Lambert W function provides the actual limit value as
where ln(z) denotes the principal branch of the complex log function. This can be shown by observing that
if c exists, so
which is the result which was to be found.
have the form
has characteristic equation , leading to and , where is the branch index. If , only need be considered.
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.
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.
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.
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.
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.
can be solved by means of the two real branches and :
This application shows in evidence that the branch difference of the W function can be employed in order to solve other trascendental equations.
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
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 .
The standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:
- An application to general relativity and quantum mechanics (quantum gravity) in lower dimensions, in fact a previously unknown link (unknown prior to) between these two areas, where the right-hand-side of (1) is now a quadratic polynomial in x:
- and where r1 and r2 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 ri and ao 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. 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. Here the right-hand-side of (1) (or (2)) is now a ratio of infinite order polynomials in x:
- where ri and si 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.
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.
The W function may be approximated using Newton's method, with successive approximations to (so ) being
The W function may also be approximated using Halley's method,
given in Corless et al. to compute W.
The LambertW function is implemented as LambertW in Maple,
lambertw in GP (and
glambertW in PARI),
lambertw in MATLAB, also
lambertw in octave with the 'specfun' package, as
lambert_w in Maxima, as
ProductLog (with a silent alias
LambertW) in Mathematica, as
lambertw in Python scipy's special function package and as
gsl_sf_lambert_Wm1 functions in special functions section of the GNU Scientific Library - GSL.
- Wright Omega function
- Lambert's trinomial equation
- Lagrange inversion theorem
- Experimental mathematics
- Holstein–Herring method
- R=T model
- Ross' π lemma
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- lambertw - MATLAB
- Maxima, a Computer Algebra System
- ProductLog at WolframAlpha
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|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
- //[strikeout] Extreme Mathematics. Monographs on the Lambert W function, its numerical approximation and generalizations for W-like inverses of transcendental forms with repeated exponential towers. //This domain does no more exist (11'2015) [/strikeout]
- GPL C++ implementation with Halley's and Fritsch's iteration.
- Special Functions of the GNU Scientific Library - GSL