In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lambert function, also called the omega function or product logarithm, is a
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
, namely the
branches
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term ''twig'' usually r ...
of the
converse relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&n ...
of the function , where is any
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
and is the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
.
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 In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.
Examples
Trigonometric inverses
Principal branches are use ...
. 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 function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
s. It is useful in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...
, for instance, in the enumeration of
trees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
. It can be used to solve various equations involving exponentials (e.g. the maxima of the
Planck,
Bose–Einstein, and
Fermi–Dirac Fermi–Dirac may refer to:
* Fermi–Dirac statistics
Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pa ...
distributions) and also occurs in the solution of
delay differential equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
DDEs are also called time ...
s, such as . In
biochemistry
Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...
, and in particular
enzyme kinetics
Enzyme kinetics is the study of the rates of enzyme-catalysed chemical reactions. In enzyme kinetics, the reaction rate is measured and the effects of varying the conditions of the reaction are investigated. Studying an enzyme's kinetics in thi ...
, an opened-form solution for the time-course kinetics analysis of
Michaelis–Menten kinetics
In biochemistry, Michaelis–Menten kinetics is one of the best-known models of enzyme kinetics. It is named after German biochemist Leonor Michaelis and Canadian physician Maud Menten. The model takes the form of an equation describing the rat ...
is described in terms of the Lambert function.
:
Terminology
The Lambert function is named after
Johann Heinrich Lambert
Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...
. The principal branch is denoted in the
Digital Library of Mathematical Functions
The Digital Library of Mathematical Functions (DLMF) is an online project at the National Institute of Standards and Technology (NIST) to develop a database of mathematical reference data for special functions and their applications. It is inte ...
, and the branch is denoted there.
The notation convention chosen here (with and ) follows the canonical reference on the Lambert function by Corless, Gonnet, Hare, Jeffrey and
Knuth.
The name "product logarithm" can be understood as this: Since the
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...
of is called the
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
, it makes sense to call the inverse "function" of the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
as "product logarithm". (Technical note: like the
complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to b ...
, it is multivalued and thus W is described as the
converse relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent&n ...
rather than inverse function.) It is related to the
Omega constant
The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
:\Omega e^\Omega = 1.
It is the value of , where is Lambert's function. The name is derived from the alternate name for Lambert's fu ...
, which is equal to .
History
Lambert first considered the related ''Lambert's Transcendental Equation'' in 1758, which led to an article by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
in 1783 that discussed the special case of .
The equation 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, 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. Prompted by this, Rob Corless and developers of the
Maple
''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since h ...
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."
Another example where this function is found is in
Michaelis–Menten kinetics
In biochemistry, Michaelis–Menten kinetics is one of the best-known models of enzyme kinetics. It is named after German biochemist Leonor Michaelis and Canadian physician Maud Menten. The model takes the form of an equation describing the rat ...
.
Although it was widely believed that the Lambert function cannot be expressed in terms of elementary (
Liouvillian) functions, the first published proof did not appear until 2008.
Elementary properties, branches and range
There are countably many branches of the function, denoted by , for integer ; being the main (or principal) 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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
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 quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is one of the oldest examples for a kinematic curve (a curve created through motion). Its discovery is attributed to the ...
, the parametric curve .
Inverse
The range plot above also delineates the regions in the complex plane where the simple inverse relationship is true. ''f'' = ''ze
z'' implies that there exists an ''n'' such that , 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 , which means that , except for where it is .
Defining , where ''x'' and ''y'' are real, and expressing ''e
z'' in polar coordinates, it is seen that
:
For
, the branch cut for is the non-positive real axis, so that
:
and
:
For
, the branch cut for is the real axis with
, so that the inequality becomes
:
Inside the regions bounded by the above, there are no discontinuous changes in , and those regions specify where the ''W'' function is simply invertible, i.e. .
Calculus
Derivative
By
implicit differentiation, one can show that all branches of satisfy the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
:
( is not
differentiable for .) 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
:
Integral
The function , and many expressions involving , can be
integrated using the
substitution
Substitution may refer to:
Arts and media
*Chord substitution, in music, swapping one chord for a related one within a chord progression
*Substitution (poetry), a variation in poetic scansion
* "Substitution" (song), a 2009 song by Silversun Pic ...
, i.e. :
:
(The last equation is more common in the literature but is undefined at ). One consequence of this (using the fact that ) is the identity
:
Asymptotic expansions
The
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of around 0 can be found using the
Lagrange inversion theorem
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.
Statement
Suppose is defined as a function of by an equ ...
and is given by
:
The
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
is , as may be seen by the
ratio test
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series
:\sum_^\infty a_n,
where each term is a real or complex number and is nonzero when is large. The test was first published by Jean le Rond d'Alembert ...
. The function defined by this series can be extended to a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...
defined on all complex numbers with a
branch cut
In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point ...
along the
interval ; this holomorphic function defines the
principal branch In mathematics, a principal branch is a function which selects one branch ("slice") of a multi-valued function. Most often, this applies to functions defined on the complex plane.
Examples
Trigonometric inverses
Principal branches are use ...
of the Lambert function.
For large values of , is asymptotic to
:
where , , and is a non-negative
Stirling number of the first kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed po ...
.
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 .
Integer and complex powers
Integer powers of also admit simple
Taylor (or
Laurent
Laurent may refer to:
*Laurent (name), a French masculine given name and a surname
**Saint Laurence (aka: Saint ''Laurent''), the martyr Laurent
**Pierre Alphonse Laurent, mathematician
**Joseph Jean Pierre Laurent, amateur astronomer, discoverer ...
) 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 .
Bounds and inequalities
A number of non-asymptotic bounds are known for the Lambert function.
Hoorfar and Hassani showed that the following bound holds for :
:
They also showed the general bound
:
for every
and
, with equality only for
.
The bound allows many other bounds to be made, such as taking
which gives the bound
:
In 2013 it was proven
that the branch can be bounded as follows:
:
:Roberto Iacono and John P. Boyd enhanced the bounds as following:
:
Identities
A few identities follow from the definition:
:
Note that, since is not
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
, it does not always hold that , much like with the
inverse trigonometric functions
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
. For fixed and , the equation has two real solutions in , one of which is of course . Then, for and , as well as for and , is the other solution.
Some other identities:
:
:
:
:
:
::(which can be extended to other and if the correct branch is chosen).
:
Substituting in the definition:
:
With Euler's iterated exponential :
:
Special values
The following are special values of the principal branch:
:
:
:
:
:
:
(the
omega constant
The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
:\Omega e^\Omega = 1.
It is the value of , where is Lambert's function. The name is derived from the alternate name for Lambert's fu ...
).
:
:
:
:
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 was found by Mező:
:
Another representation of the principal branch was found by the same author
and previously by Kalugin-Jeffrey-Corless:
:
The following
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
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
The Gaussian integral, also known as the Euler–Poisson integral, is the integral of the Gaussian function f(x) = e^ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is
\int_^\infty e^\,dx = \s ...
in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
.
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 (where the integral does not converge), 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
:
(where is an unknown real number) 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
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
.
Time dependent flow in simple branch hydraulic systems
The principal branch of the Lambert function was employed in the field of
mechanical engineering
Mechanical engineering is the study of physical machines that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, ...
, in the study of time dependent transfer of
Newtonian fluids between two reservoirs with varying free surface levels, using centrifugal pumps. The Lambert function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes:
where
is the initial flow rate and
is time.
Neuroimaging
The Lambert function was employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain
voxel
In 3D computer graphics, a voxel represents a value on a regular grid in three-dimensional space. As with pixels in a 2D bitmap, voxels themselves do not typically have their position (i.e. coordinates) explicitly encoded with their values. I ...
, to the corresponding blood oxygenation level dependent (BOLD) 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.
Crystal growth
In the crystal growth, the distribution of solute can be obtained by using the
Scheil equation
In metallurgy, the Scheil-Gulliver equation (or Scheil equation) describes solute redistribution during solidification of an alloy.
Assumptions
Four key assumptions in Scheil analysis enable determination of phases present in a cast part. ...
. So the negative principal of the Lambert W-function can be used to calculate the distribution coefficient,
:
Materials science
The Lambert function was employed in the field of
epitaxial film growth for the determination of the critical
dislocation
In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to s ...
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 (linked with the generating functions of
Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s and
Todd genus):
:
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 (also called the Jeffreys divergence ) has a closed form using the Lambert function.
Pooling of tests for infectious diseases
Solving for the optimal group size to pool tests so that at least one individual is infected involves 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
:
The Lambert function also appears in the exact solution for the bound state energy of the one dimensional Schrödinger equation with a
Double Delta Potential.
Exact solutions of the Einstein vacuum equations
In the
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...
solution of the Einstein vacuum equations, the function is needed to go from the
Eddington–Finkelstein coordinates In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (e.g. a spherically symmetric black hole) which are adapted to radial null geodesics. Null geodesics are the worldlines of pho ...
to the Schwarzschild coordinates. For this reason, it also appears in the construction of the
Kruskal–Szekeres coordinates
In general relativity, Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole. These coordinates have the advantage that they cover the entire spacet ...
.
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 (equal to ) 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 (or one if the argument of is exactly ). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.
Phase separation of polymer mixtures
In the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to the
Edmond-Ogston model, the solutions for binodal and tie-lines are formulated in terms of Lambert functions.
Wien's displacement law in a D-dimensional universe
Wien's displacement law is expressed as
. With
and
, where
is the spectral energy energy density, one finds
. The solution
shows that the spectral energy density is dependent on the dimensionality of the universe.
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
Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R, (Susceptible, Infectious, ...
, 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.
Electromagnetic surface wave propagation
The transcendental equation that appears in the determination of the propagation wave number of an electromagnetic axially symmetric surface wave (a low-attenuation single TM01 mode) propagating in a cylindrical metallic wire gives rise to an equation like (where and clump together the geometrical and physical factors of the problem), which is solved by the Lambert function. The first solution to this problem, due to Sommerfeld ''circa'' 1898, already contained an iterative method to determine the value of the Lambert function.
Orthogonal trajectories of real ellipses
The family of ellipses
centered at
is parameterized by eccentricity
. The orthogonal trajectories of this family are given by the differential equation
whose general solution is the family
.
Generalizations
The standard Lambert function expresses exact solutions to ''transcendental algebraic'' equations (in ) of the form:
where , and are real constants. The solution is
Generalizations of the Lambert function include:
- An application to
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
(quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
) in lower dimensions, in fact a link (unknown prior to 2007) between these two areas, where the right-hand side of () is replaced by a quadratic polynomial in ''x'':
where and are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which has a single argument but the terms like and are parameters of that function. In this respect, the generalization resembles the hypergeometric function and the Meijer function but it belongs to a different ''class'' of functions. When , both sides of () can be factored and reduced to () and thus the solution reduces to that of the standard function. Equation () expresses the equation governing the dilaton field, from which is derived the metric of the 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
In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
, namely the (three-dimensional) hydrogen molecule-ion
The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . It consists of two hydrogen nuclei (protons) sharing a single electron. It is the simplest molecular ion.
The ion can be formed from the ionization of a ne ...
. Here the right-hand side of () is replaced by a ratio of infinite order polynomials in :
where and are distinct real constants and is a function of the eigenenergy and the internuclear distance . Equation () with its specialized cases expressed in () and () is related to a large class of delay differential equation
In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
DDEs are also called time ...
s. G. H. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of ().
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
File:LambertWRe.png,
File:LambertWIm.png,
File:LambertWAbs.png,
File:LambertWAll.png, Superimposition of the previous three plots
Numerical evaluation
The function may be approximated using
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...
, with successive approximations to (so ) being
:
The function may also be approximated using
Halley's method
In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. It is named after its inventor Edmond Halley.
The algorithm is second in the class of Householder's m ...
,
:
given in Corless et al.
to compute .
For real
, it could be approximated by the quadratic-rate recursive formula of R. Iacono and J.P. Boyd:
:
Lajos Lóczi proves that by choosing appropriate
,
* if
:
* if
* if
** for the principal branch
:
** for the branch
:
***
for
***
for
one can determine the maximum number of iteration steps in advance for any precision:
* if
(Theorem 2.4):
* if
(Theorem 2.9):
* if
** for the principal branch
(Theorem 2.17):
** for the branch
(Theorem 2.23):
Software
The Lambert function is implemented as
LambertW
in Maple,
lambertw
in
PARI/GP, GP (and
glambertW
in
PARI),
lambertw
in
Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
, also
lambertw
in
Octave
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
with the
specfun
package, as
lambert_w
in Maxima, as
ProductLog
(with a silent alias
LambertW
) in
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiza ...
, as
lambertw
in Python
scipy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, ...
's special function package, as
LambertW
in Perl's
ntheory
module, and as
gsl_sf_lambert_W0
,
gsl_sf_lambert_Wm1
functions in th
special functionssection of th
GNU Scientific Library(GSL). In th
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.
C++ code for all the branches of the complex Lambert function is available on the homepage of István Mező.
The webpage of István Mező
/ref>
See also
* Wright Omega function
* Lambert's trinomial equation
* Lagrange inversion theorem
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.
Statement
Suppose is defined as a function of by an equ ...
* Experimental mathematics
Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with th ...
* Holstein–Herring method
* model
* Ross' lemma
Notes
References
*
*
* (Lambert function is used to solve delay-differential dynamics in human disease.)
*
*
*
Veberic, D., "Having Fun with Lambert ''W''(''x'') Function" arXiv:1003.1628 (2010)
*
External links
National Institute of Science and Technology Digital Library – Lambert
Corless et al. Notes about Lambert research
* GP
C++ implementation
with Halley's and Fritsch's iteration.
of th
GNU Scientific Library
– GSL
{{DEFAULTSORT:Lambert W Function
Special functions