Lambert W function


In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the inverse relation of the function, where is any complex number and is the exponential function.
For each integer there is one branch, denoted by, which is a complex-valued function of one complex argument. is known as the principal branch. These functions have the following property: if and are any complex numbers, then
holds if and only if
When dealing with real numbers only, the two branches and suffice: for real numbers and the equation
can be solved for only if ; we get if and the two values and if.
The Lambert 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 and also occurs in the solution of delay differential equations, such as. 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 function.

Terminology

The Lambert function is named after Johann Heinrich Lambert. The principal branch is denoted in the Digital Library of Mathematical Functions, and the branch is denoted there.
The notation convention chosen here follows the canonical reference on the Lambert function by Corless, Gonnet, Hare, Jeffrey and Knuth.

History

Lambert first considered the related Lambert's Transcendental Equation in 1758, which led to an article by Leonhard Euler in 1783 that discussed the special case of.
The function Lambert considered was
Euler transformed this equation into the form
Both authors derived a series solution for their equations.
Once Euler had solved this equation he considered the case. Taking limits he derived the equation
He then put and obtained a convergent series solution for the resulting equation, expressing x in terms of c.
After taking derivatives with respect to and some manipulation, the standard form of the Lambert function is obtained.
In 1993, when it was reported that the Lambert 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.
Another example where this function is found is in Michaelis–Menten kinetics.

Elementary properties, branches and range

There are countably many branches of the function, denoted by, for integer ; being the main branch. is defined for all complex numbers z while with is defined for all non-zero z. We have and for all.
The branch point for the principal branch is at, with a branch cut that extends to along the negative real axis. This branch cut separates the principal branch from the two branches and. In all branches with, there is a branch point at and a branch cut along the entire negative real axis.
The functions are all injective and their ranges are disjoint. The range of the entire multivalued function 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.

Inverse

The range plot above also delineates the regions in the complex plane where the simple inverse relationship W = z is true. f=zez implies that there exists an n such that z=W=W, where n will depend upon the value of z. The value of the integer n will change abruptly when zez is at the branch cut of W which will mean that zez ≤ 0, except for n=0 where it will be zez ≤ -1/e.
Define z=x+iy where x and y are real. Expressing ez in polar coordinates, it is seen that:
For, the branch cut for W will be the non-positive real axis so that:
and
For, the branch cut for W will be the real axis with so that the inequality becomes:
Inside the regions bounded by the above, there will be no discontinuous changes in W and those regions will specify where the W function is simply invertible: i.e. W = z.

Calculus

Derivative

By implicit differentiation, one can show that all branches of satisfy the differential equation
As a consequence, we get the following formula for the derivative of W:
Using the identity, we get the following equivalent formula:
At the origin we have

Antiderivative

The function, and many expressions involving, can be integrated using the substitution, i.e. :
. One consequence of this is the identity

Asymptotic expansions

The Taylor series of around 0 can be found using the Lagrange inversion theorem and is given by
The radius of convergence is, 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 ; this holomorphic function defines the principal branch of the Lambert function.
For large values of, is asymptotic to
where,, and is a non-negative Stirling number of the first kind. Keeping only the first two terms of the expansion,
The other real branch,, defined in the interval, has an approximation of the same form as approaches zero, with in this case and.
It is shown that the following bound holds :
In 2013 it was proven that the branch can be bounded as follows:

Integer and complex powers

Integer powers of also admit simple Taylor series expansions at zero:
More generally, for, the Lagrange inversion formula gives
which is, in general, a Laurent series of order. Equivalently, the latter can be written in the form of a Taylor expansion of powers of :
which holds for any and.

Identities

A few identities follow from the definition:
Note that, since is not injective, it does not always hold that, much like with the inverse trigonometric functions. For fixed and, the equation has two solutions in, one of which is of course. Then, for and, as well as for and, is the other solution.
Some other identities:
Substituting in the definition:
With Euler's iterated exponential :

Special values

For any nonzero algebraic number, is a transcendental number. Indeed, if is zero, then must be zero as well, and if is nonzero and algebraic, then by the Lindemann–Weierstrass theorem, must be transcendental, implying that must also be transcendental.
The following are special values of the principal branch:

Representations

The principal branch of the Lambert function can be represented by a proper integral, due to Poisson:
On the wider domain, the considerably simpler representation is found by Mező:
The following continued fraction representation also holds for the principal branch:
Also, if :
In turn, if, then

Other formulas

Definite integrals

There are several useful definite integral formulas involving the principal branch of the function, including the following:
The first identity can be found by writing the Gaussian integral in polar coordinates.
The second identity can be derived by making the substitution, which gives
Thus
The third identity may be derived from the second by making the substitution and the first can also be derived from the third by the substitution.
Except for along the branch cut , the principal branch of the Lambert function can be computed by the following integral:
where the two integral expressions are equivalent due to the symmetry of the integrand.

Indefinite integrals

Applications

Solving equations

The Lambert function is used to solve equations in which the unknown quantity occurs both in the base and in the exponent, or both inside and outside of a logarithm. The strategy is to convert such an equation into one of the form and then to solve for. using the function.
For example, the equation
can be solved by rewriting it as
This last equation has the desired form and the solutions for real x are:
and thus:
Generally, the solution to
is:
where a, b, and c are complex constants, with b and c not equal to zero, and the W function is of any integer order.

Viscous flows

Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in laboratory experiments can be described by using the Lambert–Euler omega function as follows:
where is the debris flow height, is the channel downstream position, is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.
In pipe flow, the Lambert W function is part of the explicit formulation of the Colebrook equation for finding the Darcy friction factor. This factor is used to determine the pressure drop through a straight run of pipe when the flow is turbulent.

Neuroimaging

The Lambert 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 signal.

Chemical engineering

The Lambert 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 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.

Materials science

The Lambert 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 for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert turns it in an explicit equation for analytical handling with ease.

Porous media

The Lambert 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 homogeneous 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 lighter fluid.

Bernoulli numbers and Todd genus

The equation :
can be solved by means of the two real branches and :
This application shows that the branch difference of the function can be employed in order to solve other transcendental equations.

Statistics

The centroid of a set of histograms defined with respect to the symmetrized Kullback–Leibler divergence has a closed form using the Lambert function.

Exact solutions of the Schrödinger equation

The Lambert function appears in a quantum-mechanical 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

Exact solutions of the Einstein vacuum equations

In the Schwarzschild metric solution of the Einstein vacuum equations, the function is needed to go from the Eddington–Finkelstein coordinates to the Schwarzschild coordinates. For this reason, it also appears in the construction of the Kruskal–Szekeres coordinates.

Resonances of the delta-shell potential

The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert function.

Thermodynamic equilibrium

If a reaction involves reactants and products having heat capacities that are constant with temperature then the equilibrium constant obeys
for some constants,, and. When is not zero we can find the value or values of where equals a given value as follows, where we use for.
If and have the same sign there will be either two solutions or none. If they have opposite signs, there will be one solution.

AdS/CFT correspondence

The classical finite-size corrections to the dispersion relations of giant magnons, single spikes and GKP strings can be expressed in terms of the Lambert function.

Epidemiology

In the limit of the SIR model, the proportion of susceptible and recovered individuals has a solution in terms of the Lambert function.

Determination of the time of flight of a projectile

The total time of the journey of a projectile which experiences air resistance proportional to its velocity can be determined in exact form by using the Lambert function.

Generalizations

The standard Lambert function expresses exact solutions to transcendental algebraic equations of the form:
where, and are real constants. The solution is
Generalizations of the Lambert function include:
Applications of the Lambert function in fundamental physical problems are not exhausted even for the standard case expressed in as seen recently in the area of atomic, molecular, and optical physics.

Plots

Numerical evaluation

The function may be approximated using Newton's method, with successive approximations to being
The function may also be approximated using Halley's method,
given in Corless et al. to compute.

Software

The Lambert function is implemented as
, lambertw in GP, lambertw in Matlab, also lambertw in octave with the specfun package, as lambert_w in Maxima, as ProductLog in Mathematica, as lambertw in Python scipy's special function package, as LambertW in Perl's ntheory module, and as gsl_sf_lambert_W0, gsl_sf_lambert_Wm1 functions in the section of the . In the , the calls are lambert_w0, lambert_wm1, lambert_w0_prime, and lambert_wm1_prime. In R, the Lambert function is implemented as the lambertW0 and lambertWm1 functions in the lamW package.
A C++ code for all the branches of the complex Lambert function is available on the homepage of István Mező.