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Lambert W function

In mathematics, the Lambert W function, also called the omega function or product logarithm,[1] is a multivalued function, namely the branches of the converse relation of the function f(w) = wew, where w is any complex number and ew is the exponential function.

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

The graph of y = W(x) for real x < 6 and y > −4. The upper branch (blue) with y ≥ −1 is the graph of the function W0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W−1. The minimum value of x is at {−1/e,−1}

For each integer k there is one branch, denoted by Wk(z), which is a complex-valued function of one complex argument. W0 is known as the principal branch. These functions have the following property: if z and w are any complex numbers, then

holds if and only if

When dealing with real numbers only, the two branches W0 and W−1 suffice: for real numbers x and y the equation

can be solved for y only if x ≥ −1/e; we get y = W0(x) if x ≥ 0 and the two values y = W0(x) and y = W−1(x) if 1/ex < 0.

The Lambert W relation cannot be expressed in terms of elementary functions.[2] 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, an opened-form solution for the time-course kinetics analysis of Michaelis–Menten kinetics is described in terms of the Lambert W function.

Main branch of the Lambert W function in the complex plane, plotted with domain coloring. Note the branch cut along the negative real axis, ending at 1/e.
The modulus of the principal branch of the Lambert W function, colored according to arg W(z)

Terminology

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) = ew is called the logarithm, it makes sense to call the inverse "function" of the product wew 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).

History

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 wew.

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 a = b. Taking limits, he derived the equation

 

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]

Elementary properties, branches and range

 
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 Wk(z), for integer k; W0(z) being the main (or principal) branch. W0(z) is defined for all complex numbers z while Wk(z) with k ≠ 0 is defined for all non-zero z. We have W0(0) = 0 and limz→0 Wk(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 Wk with k ≠ 0, there is a branch point at z = 0 and a branch cut along the entire negative real axis.

The functions Wk(z), kZ 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  , 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   (blue) region and the   (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   is true. f = zez 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 zez is at the branch cut of  , which means that zez ≤ 0, except for   where it is zez ≤ −1/e.

Defining  , where x and y are real, and expressing ez 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 W satisfy the differential equation

 

(W is not differentiable for z = −1/e.) As a consequence, we get the following formula for the derivative of W:

 

Using the identity eW(z) = z/W(z), we get the following equivalent formula:

 

At the origin we have

 

Integral

The function W(x), and many expressions involving W(x), can be integrated using the substitution w = W(x), i.e. x = wew:

 

(The last equation is more common in the literature but is undefined at x = 0). One consequence of this (using the fact that W0(e) = 1) is the identity

 

Asymptotic expansions

The Taylor series of W0 around 0 can be found using the Lagrange inversion theorem and is given by

 

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

 

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,

 

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]

Integer and complex powers

Integer powers of W0 also admit simple Taylor (or Laurent) series expansions at zero:

 

More generally, for rZ, 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 W0(x) / x:

 

which holds for any rC and |x| < 1/e.

Bounds and inequalities

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

Hoorfar and Hassani[10] showed that the following bound holds for xe:

 

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[11] that the branch W−1 can be bounded as follows:

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

Identities

 
A plot of Wj(x ex) where blue is for j=0 and red is for j=−1. The diagonal line represents the intervals where Wj(x ex)=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:

 

Note that, since f(x) = xex 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 xex = yey 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 = Wi(xex) is the other solution.

Some other identities:[13]

 
 [14]
 
 
 
(which can be extended to other n and x if the correct branch is chosen).
 

Substituting −ln x in the definition:[15]

 

With Euler's iterated exponential h(x):

 

Special values

The following are special values of the principal branch:

 
 
 
 
 
  (the omega constant).
 
 
 
 

Representations

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

 

On the wider domain 1/exe, the considerably simpler representation was found by Mező:[17]

 

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

 

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

 

Also, if |W(x)| < 1:[21]

 

In turn, if |W(x)| > e, then

 

Other formulas

Definite integrals

There are several useful definite integral formulas involving the principal branch of the W 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 u = W(x), which gives

 

Thus

 

The third identity may be derived from the second by making the substitution u = x−2 and the first can also be derived from the third by the substitution z = 1/2 tan x.

Except for z along the branch cut (−∞, −1/e] (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:[22]

 

where the two integral expressions are equivalent due to the symmetry of the integrand.

Indefinite integrals

 
1st proof

Introduce substitution variable  

 
 
 
 
 
 
2nd proof

 

 

 

 

 

 

 

 

 

 
Proof

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
Proof

Introduce substitution variable  , which gives us   and  

 

 

 

 

 

 

 

 

Applications

Solving equations

The Lambert W 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 zez = w and then to solve for z using the W function.

For example, the equation

 

(where x 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 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.

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.[23]

Time dependent flow in simple branch hydraulic systems

The principal branch of the Lambert W function was employed in the field of mechanical engineering, in the study of time dependent transfer of Newtonian fluids between two reservoirs with varying free surface levels, using centrifugal pumps.[24] The Lambert W 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 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.[25]

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.[26][27]

Crystal growth

In the crystal growth, the distribution of solute can be obtained by using the Scheil equation. So the negative principal of the Lambert W-function can be used to calculate the distribution coefficient,  :[28]

 

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.[29]

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 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.[30]

Bernoulli numbers and Todd genus

The equation (linked with the generating functions of Bernoulli numbers and Todd genus):

 

can be solved by means of the two real branches W0 and W−1:

 

This application shows that the branch difference of the W function can be employed in order to solve other transcendental equations.[31]

Statistics

The centroid of a set of histograms defined with respect to the symmetrized Kullback–Leibler divergence (also called the Jeffreys divergence [32]) has a closed form using the Lambert W function.[33]

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 W function.[34][35][36]

Exact solutions of the Schrödinger equation

The Lambert W 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[37]

 

The Lambert W 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 solution of the Einstein vacuum equations, the W 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 W function.[38]

Thermodynamic equilibrium

If a reaction involves reactants and products having heat capacities that are constant with temperature then the equilibrium constant K obeys

 

for some constants a, b, and c. When c (equal to ΔCp/R) is not zero we can find the value or values of T where K equals a given value as follows, where we use L for ln T.

 

If a and c have the same sign there will be either two solutions or none (or one if the argument of W is exactly 1/e). (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 W functions.[39]

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.[40]

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 W function.[41][42]

Epidemiology

In the t → ∞ limit of the SIR model, the proportion of susceptible and recovered individuals has a solution in terms of the Lambert W function.[43]

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 W 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 u ln u = v (where u and v clump together the geometrical and physical factors of the problem), which is solved by the Lambert W function. The first solution to this problem, due to Sommerfeld circa 1898, already contained an iterative method to determine the value of the Lambert W function.[44]

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 W function expresses exact solutions to transcendental algebraic equations (in x) of the form:

 

 

 

 

 

(1)

where a0, c and r are real constants. The solution is

 
Generalizations of the Lambert W function[45][46][47] include:
  • An application to general relativity and quantum mechanics (quantum gravity) in lower dimensions, in fact a link (unknown prior to 2007[48]) between these two areas, where the right-hand side of (1) is replaced by a quadratic polynomial in x:
     

     

     

     

     

    (2)

    where r1 and r2 are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function which has a single argument x but the terms like ri and a0 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. Equation (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.[49] Here the right-hand side of (1) is replaced by a ratio of infinite order polynomials in x:
     

     

     

     

     

    (3)

    where ri and si are distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Equation (3) with its specialized cases expressed in (1) and (2) is related to a large class of delay differential equations. G. H. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of (3).[50]

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.[51]

Plots

Numerical evaluation

The W function may be approximated using Newton's method, with successive approximations to w = W(z) (so z = wew) being

 

The W function may also be approximated using Halley's method,

 

given in Corless et al.[3] to compute W.


For real  , it could be approximated by the quadratic-rate recursive formula of R. Iacono and J.P. Boyd:[52]

 

Lajos Lóczi proves that by choosing appropriate  ,

  • if  :  
  • if    
  • if  
    • for the principal branch  :  
    • for the branch  :
      •   for  
      •   for
lambert, function, mathematics, also, called, omega, function, product, logarithm, multivalued, function, namely, branches, converse, relation, function, where, complex, number, exponential, function, product, logarithm, plotted, complex, plane, from, graph, r. In mathematics the Lambert W function also called the omega function or product logarithm 1 is a multivalued function namely the branches of the converse relation of the function f w wew where w is any complex number and ew is the exponential function The product logarithm Lambert W function plotted in the complex plane from 2 2i to 2 2i The graph of y W x for real x lt 6 and y gt 4 The upper branch blue with y 1 is the graph of the function W0 principal branch the lower branch magenta with y 1 is the graph of the function W 1 The minimum value of x is at 1 e 1 For each integer k there is one branch denoted by Wk z which is a complex valued function of one complex argument W0 is known as the principal branch These functions have the following property if z and w are any complex numbers then w e w z displaystyle we w z holds if and only if w W k z for some integer k displaystyle w W k z text for some integer k When dealing with real numbers only the two branches W0 and W 1 suffice for real numbers x and y the equation y e y x displaystyle ye y x can be solved for y only if x 1 e we get y W0 x if x 0 and the two values y W0 x and y W 1 x if 1 e x lt 0 The Lambert W relation cannot be expressed in terms of elementary functions 2 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 an opened form solution for the time course kinetics analysis of Michaelis Menten kinetics is described in terms of the Lambert W function Main branch of the Lambert W function in the complex plane plotted with domain coloring Note the branch cut along the negative real axis ending at 1 e The modulus of the principal branch of the Lambert W function colored according to arg W z Contents 1 Terminology 2 History 3 Elementary properties branches and range 3 1 Inverse 4 Calculus 4 1 Derivative 4 2 Integral 5 Asymptotic expansions 5 1 Integer and complex powers 6 Bounds and inequalities 7 Identities 8 Special values 9 Representations 10 Other formulas 10 1 Definite integrals 10 2 Indefinite integrals 11 Applications 11 1 Solving equations 11 2 Viscous flows 11 3 Time dependent flow in simple branch hydraulic systems 11 4 Neuroimaging 11 5 Chemical engineering 11 6 Crystal growth 11 7 Materials science 11 8 Porous media 11 9 Bernoulli numbers and Todd genus 11 10 Statistics 11 11 Pooling of tests for infectious diseases 11 12 Exact solutions of the Schrodinger equation 11 13 Exact solutions of the Einstein vacuum equations 11 14 Resonances of the delta shell potential 11 15 Thermodynamic equilibrium 11 16 Phase separation of polymer mixtures 11 17 Wien s displacement law in a D dimensional universe 11 18 AdS CFT correspondence 11 19 Epidemiology 11 20 Determination of the time of flight of a projectile 11 21 Electromagnetic surface wave propagation 12 Generalizations 13 Plots 14 Numerical evaluation 15 Software 16 See also 17 Notes 18 References 19 External linksTerminology EditThe 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 ew is called the logarithm it makes sense to call the inverse function of the product wew 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 History EditLambert 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 wew The equation Lambert considered was x x m q displaystyle x x m q Euler transformed this equation into the form x a x b a b c x a b 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 ln x c x a 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 Elementary properties branches and range Edit 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 Wk z for integer k W0 z being the main or principal branch W0 z is defined for all complex numbers z while Wk z with k 0 is defined for all non zero z We have W0 0 0 and lim z 0 Wk 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 Wk with k 0 there is a branch point at z 0 and a branch cut along the entire negative real axis The functions Wk 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 Edit Regions of the complex plane for which W n z e z z 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 n 1 displaystyle n 1 blue region and the n 0 displaystyle n 0 gray region Horizontal grid lines are in multiples of p The range plot above also delineates the regions in the complex plane where the simple inverse relationship W n z e z z displaystyle W n ze z z is true f zez implies that there exists an n such that z W n f W n z e z 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 zez is at the branch cut of W n z e z displaystyle W n ze z which means that zez 0 except for n 0 displaystyle n 0 where it is zez 1 e Defining z x i y displaystyle z x iy where x and y are real and expressing ez in polar coordinates it is seen that z e z x i y e x cos y i sin y e x x cos y y sin y i e x x sin y y cos y displaystyle begin aligned ze z amp x iy e x cos y i sin y amp e x x cos y y sin y ie x x sin y y cos y end aligned For n 0 displaystyle n neq 0 the branch cut for W n z e z displaystyle W n ze z is the non positive real axis so that x sin y y cos y 0 x y tan y displaystyle x sin y y cos y 0 Rightarrow x y tan y and x cos y y sin y e x 0 displaystyle x cos y y sin y e x leq 0 For n 0 displaystyle n 0 the branch cut for W n z e z displaystyle W n ze z is the real axis with lt z 1 e displaystyle infty lt z leq 1 e so that the inequality becomes x cos y y sin y e x 1 e 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 W n z e z displaystyle W n ze z and those regions specify where the W function is simply invertible i e W n z e z z displaystyle W n ze z z Calculus EditDerivative Edit By implicit differentiation one can show that all branches of W satisfy the differential equation z 1 W d W d z W for z 1 e displaystyle z 1 W frac dW dz W quad text for z neq frac 1 e W is not differentiable for z 1 e As a consequence we get the following formula for the derivative of W d W d z W z z 1 W z for z 0 1 e displaystyle frac dW dz frac W z z 1 W z quad text for z not in left 0 frac 1 e right Using the identity eW z z W z we get the following equivalent formula d W d z 1 z e W z for z 1 e displaystyle frac dW dz frac 1 z e W z quad text for z neq frac 1 e At the origin we have W 0 0 1 displaystyle W 0 0 1 Integral Edit The function W x and many expressions involving W x can be integrated using the substitution w W x i e x wew W x d x x W x x e W x C x W x 1 1 W x C displaystyle begin aligned int W x dx amp xW x x e W x C amp x left W x 1 frac 1 W x right C end aligned The last equation is more common in the literature but is undefined at x 0 One consequence of this using the fact that W0 e 1 is the identity 0 e W 0 x d x e 1 displaystyle int 0 e W 0 x dx e 1 Asymptotic expansions EditThe Taylor series of W0 around 0 can be found using the Lagrange inversion theorem and is given by W 0 x n 1 n n 1 n x n x x 2 3 2 x 3 8 3 x 4 125 24 x 5 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 W 0 x L 1 L 2 L 2 L 1 L 2 2 L 2 2 L 1 2 L 2 6 9 L 2 2 L 2 2 6 L 1 3 L 2 12 36 L 2 22 L 2 2 3 L 2 3 12 L 1 4 L 1 L 2 l 0 m 1 1 l l m l 1 m L 1 l m L 2 m displaystyle begin aligned W 0 x amp 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 amp 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 ml 1 is a non negative Stirling number of the first kind 3 Keeping only the first two terms of the expansion W 0 x ln x ln ln x o 1 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 Integer and complex powers Edit Integer powers of W0 also admit simple Taylor or Laurent series expansions at zero W 0 x 2 n 2 2 n n 3 n 2 x n x 2 2 x 3 4 x 4 25 3 x 5 18 x 6 displaystyle W 0 x 2 sum n 2 infty frac 2 left n right n 3 n 2 x n x 2 2x 3 4x 4 tfrac 25 3 x 5 18x 6 cdots More generally for r Z the Lagrange inversion formula gives W 0 x r n r r n n r 1 n r x n displaystyle W 0 x r sum n r infty frac r left n right n r 1 n r x n 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 W0 x x W 0 x x r e r W 0 x n 0 r n r n 1 n x n displaystyle left frac W 0 x x right r e rW 0 x sum n 0 infty frac r left n r right n 1 n left x right n which holds for any r C and x lt 1 e Bounds and inequalities EditA number of non asymptotic bounds are known for the Lambert function Hoorfar and Hassani 10 showed that the following bound holds for x e ln x ln ln x ln ln x 2 ln x W 0 x ln x ln ln x e e 1 ln ln x ln x 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 W 0 x log x y 1 log y displaystyle W 0 x leq log left frac x y 1 log y right for every y gt 1 e displaystyle y gt 1 e and x 1 e displaystyle x geq 1 e with equality only for x y log y displaystyle x y log y The bound allows many other bounds to be made such as taking y x 1 displaystyle y x 1 which gives the bound W 0 x log 2 x 1 1 log x 1 displaystyle W 0 x leq log left frac 2x 1 1 log x 1 right In 2013 it was proven 11 that the branch W 1 can be bounded as follows 1 2 u u lt W 1 e u 1 lt 1 2 u 2 3 u for u gt 0 displaystyle 1 sqrt 2u u lt W 1 left e u 1 right lt 1 sqrt 2u tfrac 2 3 u quad text for u gt 0 Roberto Iacono and John P Boyd 12 enhanced the bounds as following ln x ln x ln x ln x 1 ln x ln x ln 1 ln ln x ln x W 0 x ln x ln x ln 1 ln ln x ln x 1 ln 1 ln ln x ln x 1 ln x ln x displaystyle ln frac x ln x frac ln frac x ln x 1 ln frac x ln x ln 1 frac ln ln x ln x leq W 0 x leq ln frac x ln x ln 1 frac ln ln x ln x 1 frac ln 1 frac ln ln x ln x 1 ln frac x ln x Identities Edit A plot of Wj x ex where blue is for j 0 and red is for j 1 The diagonal line represents the intervals where Wj x ex 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 W 0 x e x x for x 1 W 1 x e x x for x 1 displaystyle begin aligned W 0 xe x amp x amp text for x amp geq 1 W 1 xe x amp x amp text for x amp leq 1 end aligned Note that since f x xex is not injective it does not always hold that W f x x much like with the inverse trigonometric functions For fixed x lt 0 and x 1 the equation xex yey has two real solutions in y one of which is of course y x Then for i 0 and x lt 1 as well as for i 1 and x 1 0 y Wi xex is the other solution Some other identities 13 W x e W x x therefore e W x x W x e W x W x x e n W x x W x n displaystyle begin aligned amp W x e W x x quad text therefore 5pt amp 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 ln W 0 x ln x W 0 x for x gt 0 displaystyle ln W 0 x ln x W 0 x quad text for x gt 0 14 W 0 x ln x ln x and e W 0 x ln x x for 1 e x 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 W 1 x ln x ln x and e W 1 x ln x x for 0 lt x 1 e 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 lt x leq frac 1 e W x ln x W x for x 1 e W n x n W x n 1 n W x for n x gt 0 displaystyle begin aligned amp W x ln frac x W x amp amp text for x geq frac 1 e 5pt amp W left frac nx n W left x right n 1 right nW x amp amp text for n x gt 0 end aligned which can be extended to other n and x if the correct branch is chosen dd W x W y W x y 1 W x 1 W y for x y gt 0 displaystyle W x W y W left xy left frac 1 W x frac 1 W y right right quad text for x y gt 0 Substituting ln x in the definition 15 W 0 ln x x ln x for 0 lt x e W 1 ln x x ln x for x gt e displaystyle begin aligned W 0 left frac ln x x right amp ln x amp text for 0 amp lt x leq e 5pt W 1 left frac ln x x right amp ln x amp text for x amp gt e end aligned With Euler s iterated exponential h x h x e W ln x W ln x ln x for x 1 displaystyle begin aligned h x amp e W ln x amp frac W ln x ln x quad text for x neq 1 end aligned Special values EditThe following are special values of the principal branch W p 2 i p 2 displaystyle W left frac pi 2 right frac i pi 2 W 1 e 1 displaystyle W left frac 1 e right 1 W 2 ln 2 ln 2 displaystyle W left 2 ln 2 right ln 2 W x ln x ln x provided x 1 e 0 36788 displaystyle W left x ln x right ln x text provided x geqslant 1 e approx 0 36788 W 0 0 displaystyle W 0 0 W 1 W d t e t t 2 p 2 1 1 0 56714329 displaystyle W 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 W 1 e W 1 ln 1 W 1 ln W 1 displaystyle W 1 e W 1 ln left frac 1 W 1 right ln W 1 W e 1 displaystyle W e 1 W e 1 e e displaystyle W left e 1 e right e W 1 0 31813 1 33723 i displaystyle W 1 approx 0 31813 1 33723i Representations EditThe principal branch of the Lambert function can be represented by a proper integral due to Poisson 16 p 2 W x 0 p sin 3 2 t x e cos t sin 5 2 t sin t 1 2 x e cos t cos t sin t x 2 e 2 cos t sin 1 2 t d t for x lt 1 e displaystyle frac pi 2 W 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 lt frac 1 e On the wider domain 1 e x e the considerably simpler representation was found by Mezo 17 W x 1 p Re 0 p ln e e i t x e i t e e i t x e i t d t displaystyle W 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 W x 1 p 0 p log 1 x sin t t e t cot t d t displaystyle W x frac 1 pi int 0 pi log 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 W x x 1 x 1 x 2 5 x 3 17 x 10 133 x 17 1927 x 190 13582711 x 94423 displaystyle W 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 W x lt 1 21 W x x exp x exp x displaystyle W x cfrac x exp cfrac x exp cfrac x ddots In turn if W x gt e then W x ln x ln x ln x displaystyle W x ln cfrac x ln cfrac x ln cfrac x ddots Other formulas EditDefinite integrals Edit There are several useful definite integral formulas involving the principal branch of the W function including the following 0 p W 2 cot 2 x sec 2 x d x 4 p 0 W x x x d x 2 2 p 0 W 1 x 2 d x 2 p displaystyle begin aligned amp int 0 pi W left 2 cot 2 x right sec 2 x dx 4 sqrt pi 5pt amp int 0 infty frac W x x sqrt x dx 2 sqrt 2 pi 5pt amp int 0 infty W left frac 1 x 2 right dx sqrt 2 pi end aligned The first identity can be found by writing the Gaussian integral in polar coordinates The second identity can be derived by making the substitution u W x which gives x u e u d x d u u 1 e u displaystyle begin aligned x amp ue u 5pt frac dx du amp u 1 e u end aligned Thus 0 W x x x d x 0 u u e u u e u u 1 e u d u 0 u 1 u e u d u 0 u 1 u 1 e u d u 0 u 1 2 e u 2 d u 0 u 1 2 e u 2 d u 2 0 2 w 1 2 e w d w 2 0 2 w 1 2 e w d w u 2 w 2 2 0 w 1 2 e w d w 2 0 w 1 2 e w d w 2 2 G 3 2 2 G 1 2 2 2 1 2 p 2 p 2 2 p displaystyle begin aligned int 0 infty frac W x x sqrt x dx amp int 0 infty frac u ue u sqrt ue u u 1 e u du 5pt amp int 0 infty frac u 1 sqrt ue u du 5pt amp int 0 infty frac u 1 sqrt u frac 1 sqrt e u du 5pt amp int 0 infty u tfrac 1 2 e frac u 2 du int 0 infty u tfrac 1 2 e frac u 2 du 5pt amp 2 int 0 infty 2w tfrac 1 2 e w dw 2 int 0 infty 2w tfrac 1 2 e w dw amp amp quad u 2w 5pt amp 2 sqrt 2 int 0 infty w tfrac 1 2 e w dw sqrt 2 int 0 infty w tfrac 1 2 e w dw 5pt amp 2 sqrt 2 cdot Gamma left tfrac 3 2 right sqrt 2 cdot Gamma left tfrac 1 2 right 5pt amp 2 sqrt 2 left tfrac 1 2 sqrt pi right sqrt 2 left sqrt pi right 5pt amp 2 sqrt 2 pi end aligned The third identity may be derived from the second by making the substitution u x 2 and the first can also be derived from the third by the substitution z 1 2 tan x Except for z along the branch cut 1 e where the integral does not converge the principal branch of the Lambert W function can be computed by the following integral 22 W z z 2 p p p 1 n cot n 2 n 2 z n csc n e n cot n d n z p 0 p 1 n cot n 2 n 2 z n csc n e n cot n d n displaystyle begin aligned W z amp frac z 2 pi int pi pi frac left 1 nu cot nu right 2 nu 2 z nu csc nu e nu cot nu d nu 5pt amp frac z pi int 0 pi frac left 1 nu cot nu right 2 nu 2 z nu csc nu e nu cot nu d nu end aligned where the two integral expressions are equivalent due to the symmetry of the integrand Indefinite integrals Edit W x x d x W x 2 2 W x C displaystyle int frac W x x dx frac W x 2 2 W x C 1st proof Introduce substitution variable u W x u e u x d d u u e u u 1 e u displaystyle u W x rightarrow ue u x frac d du ue u u 1 e u W x x d x u u e u u 1 e u d u displaystyle int frac W x x dx int frac u ue u u 1 e u du W x x d x u u e u u 1 e u d u displaystyle int frac W x x dx int frac cancel color OliveGreen u cancel color OliveGreen u cancel color BrickRed e u left u 1 right cancel color BrickRed e u du W x x d x u 1 d u displaystyle int frac W x x dx int u 1 du W x x d x u 2 2 u C displaystyle int frac W x x dx frac u 2 2 u C u W x displaystyle u W x dd W x x d x W x 2 2 W x C displaystyle int frac W x x dx frac W x 2 2 W x C 2nd proof W x e W x x W x x e W x displaystyle W x e W x x rightarrow frac W x x e W x W x x d x e W x d x displaystyle int frac W x x dx int e W x dx u W x u e u x d d u u e u u 1 e u displaystyle u W x rightarrow ue u x frac d du ue u left u 1 right e u dd W x x d x e u u 1 e u d u displaystyle int frac W x x dx int e u u 1 e u du W x x d x e u u 1 e u d u displaystyle int frac W x x dx int cancel color OliveGreen e u left u 1 right cancel color OliveGreen e u du W x x d x u 1 d u displaystyle int frac W x x dx int u 1 du W x x d x u 2 2 u C displaystyle int frac W x x dx frac u 2 2 u C u W x displaystyle u W x dd W x x d x W x 2 2 W x C displaystyle int frac W x x dx frac W x 2 2 W x C W A e B x d x W A e B x 2 2 B W A e B x B C displaystyle int W left Ae Bx right dx frac W left Ae Bx right 2 2B frac W left Ae Bx right B C Proof W A e B x d x W A e B x d x displaystyle int W left Ae Bx right dx int W left Ae Bx right dx u B x u B x d d u u B 1 B displaystyle u Bx rightarrow frac u B x frac d du frac u B frac 1 B dd W A e B x d x W A e u 1 B d u displaystyle int W left Ae Bx right dx int W left Ae u right frac 1 B du v e u ln v u d d v ln v 1 v displaystyle v e u rightarrow ln left v right u frac d dv ln left v right frac 1 v dd W A e B x d x 1 B W A v v d v displaystyle int W left Ae Bx right dx frac 1 B int frac W left Av right v dv w A v w A v d d w w A 1 A displaystyle w Av rightarrow frac w A v frac d dw frac w A frac 1 A dd W A e B x d x 1 B A W w w 1 A d w displaystyle int W left Ae Bx right dx frac 1 B int frac cancel color OliveGreen A W w w cancel color OliveGreen frac 1 A dw t W w t e t w d d t t e t t 1 e t displaystyle t W left w right rightarrow te t w frac d dt te t left t 1 right e t dd W A e B x d x 1 B t t e t t 1 e t d t displaystyle int W left Ae Bx right dx frac 1 B int frac t te t left t 1 right e t dt W A e B x d x 1 B t t e t t 1 e t d t displaystyle int W left Ae Bx right dx frac 1 B int frac cancel color OliveGreen t cancel color OliveGreen t cancel color BrickRed e t left t 1 right cancel color BrickRed e t dt W A e B x d x 1 B t 1 d t displaystyle int W left Ae Bx right dx frac 1 B int t 1dt W A e B x d x t 2 2 B t B C displaystyle int W left Ae Bx right dx frac t 2 2B frac t B C t W w displaystyle t W left w right dd W A e B x d x W w 2 2 B W w B C displaystyle int W left Ae Bx right dx frac W left w right 2 2B frac W left w right B C w A v displaystyle w Av dd W A e B x d x W A v 2 2 B W A v B C displaystyle int W left Ae Bx right dx frac W left Av right 2 2B frac W left Av right B C v e u displaystyle v e u dd W A e B x d x W A e u 2 2 B W A e u B C displaystyle int W left Ae Bx right dx frac W left Ae u right 2 2B frac W left Ae u right B C u B x displaystyle u Bx dd W A e B x d x W A e B x 2 2 B W A e B x B C displaystyle int W left Ae Bx right dx frac W left Ae Bx right 2 2B frac W left Ae Bx right B C W x x 2 d x Ei W x e W x C displaystyle int frac W x x 2 dx operatorname Ei left W x right e W x C Proof Introduce substitution variable u W x displaystyle u W x which gives us u e u x displaystyle ue u x and d d u u e u u 1 e u displaystyle frac d du ue u left u 1 right e u W x x 2 d x u u e u 2 u 1 e u d u u 1 u e u d u u u e u d u 1 u e u d u e u d u e u u d u displaystyle begin aligned int frac W x x 2 dx amp int frac u left ue u right 2 left u 1 right e u du amp int frac u 1 ue u du amp int frac u ue u du int frac 1 ue u du amp int e u du int frac e u u du end aligned v u v u d d v v 1 displaystyle v u rightarrow v u frac d dv v 1 dd W x x 2 d x e v 1 d v e u u d u displaystyle int frac W x x 2 dx int e v left 1 right dv int frac e u u du W x x 2 d x e v Ei u C displaystyle int frac W x x 2 dx e v operatorname Ei left u right C v u displaystyle v u dd W x x 2 d x e u Ei u C displaystyle int frac W x x 2 dx e u operatorname Ei left u right C u W x displaystyle u W x dd W x x 2 d x e W x Ei W x C Ei W x e W x C displaystyle begin aligned int frac W x x 2 dx amp e W x operatorname Ei left W x right C amp operatorname Ei left W x right e W x C end aligned Applications EditSolving equations Edit The Lambert W 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 zez w and then to solve for z using the W function For example the equation 3 x 2 x 2 displaystyle 3 x 2x 2 where x is an unknown real number can be solved by rewriting it as x 1 3 x 1 2 multiply by 3 x 2 x 1 3 x 1 1 6 multiply by 1 3 ln 3 x 1 e ln 3 x 1 ln 3 6 multiply by ln 3 displaystyle begin aligned amp x 1 3 x frac 1 2 amp mbox multiply by 3 x 2 Leftrightarrow amp x 1 3 x 1 frac 1 6 amp mbox multiply by 1 3 Leftrightarrow amp ln 3 x 1 e ln 3 x 1 frac ln 3 6 amp mbox multiply by ln 3 end aligned This last equation has the desired form and the solutions for real x are ln 3 x 1 W 0 ln 3 6 or ln 3 x 1 W 1 ln 3 6 displaystyle ln 3 x 1 W 0 left frac ln 3 6 right textrm or ln 3 x 1 W 1 left frac ln 3 6 right and thus x 1 W 0 ln 3 6 ln 3 0 79011 or x 1 W 1 ln 3 6 ln 3 1 44456 displaystyle x 1 frac W 0 left frac ln 3 6 right ln 3 0 79011 ldots textrm or x 1 frac W 1 left frac ln 3 6 right ln 3 1 44456 ldots Generally the solution to x a b e c x displaystyle x a b e cx is x a 1 c W b c e a c displaystyle x a frac 1 c W bc e ac 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 Edit 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 H x 1 W H 0 1 e H 0 1 x L displaystyle H x 1 W left H 0 1 e H 0 1 frac x L right 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 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 23 Time dependent flow in simple branch hydraulic systems Edit The principal branch of the Lambert W function was employed in the field of mechanical engineering in the study of time dependent transfer of Newtonian fluids between two reservoirs with varying free surface levels using centrifugal pumps 24 The Lambert W function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes Q turb Q i z i W 0 z i e z i b t b Q lam Q i 3 i W 0 3 i e 3 i b t b G 1 displaystyle begin aligned Q text turb amp frac Q i zeta i W 0 left zeta i e zeta i beta t b right Q text lam amp frac Q i xi i W 0 left xi i e left xi i beta t b Gamma 1 right right end aligned where Q i displaystyle Q i is the initial flow rate and t displaystyle t is time Neuroimaging Edit 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 25 Chemical engineering Edit 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 26 27 Crystal growth Edit In the crystal growth the distribution of solute can be obtained by using the Scheil equation So the negative principal of the Lambert W function can be used to calculate the distribution coefficient k textstyle k 28 k W Z ln 1 f s Z C S C 0 1 f s ln 1 f s displaystyle begin aligned amp k frac W Z ln 1 fs amp Z frac C S C 0 1 fs ln 1 fs end aligned Materials science Edit 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 29 Porous media Edit 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 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 30 Bernoulli numbers and Todd genus Edit The equation linked with the generating functions of Bernoulli numbers and Todd genus Y X 1 e X displaystyle Y frac X 1 e X can be solved by means of the two real branches W0 and W 1 X Y W 1 Y e Y W 0 Y e Y Y W 0 Y e Y for Y lt 1 W 0 Y e Y W 1 Y e Y Y W 1 Y e Y for 1 lt Y lt 0 displaystyle X Y begin cases W 1 left Ye Y right W 0 left Ye Y right Y W 0 left Ye Y right amp text for Y lt 1 W 0 left Ye Y right W 1 left Ye Y right Y W 1 left Ye Y right amp text for 1 lt Y lt 0 end cases This application shows that the branch difference of the W function can be employed in order to solve other transcendental equations 31 Statistics Edit The centroid of a set of histograms defined with respect to the symmetrized Kullback Leibler divergence also called the Jeffreys divergence 32 has a closed form using the Lambert W function 33 Pooling of tests for infectious diseases Edit Solving for the optimal group size to pool tests so that at least one individual is infected involves the Lambert W function 34 35 36 Exact solutions of the Schrodinger equation Edit The Lambert W 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 Schrodinger equation in terms of the confluent hypergeometric functions The potential is given as V V 0 1 W e x s displaystyle V frac V 0 1 W left e frac x sigma right A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrodinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to 37 z W e x s displaystyle z W left e frac x sigma right The Lambert W function also appears in the exact solution for the bound state energy of the one dimensional Schrodinger equation with a Double Delta Potential Exact solutions of the Einstein vacuum equations Edit In the Schwarzschild metric solution of the Einstein vacuum equations the W 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 Edit The s wave resonances of the delta shell potential can be written exactly in terms of the Lambert W function 38 Thermodynamic equilibrium Edit If a reaction involves reactants and products having heat capacities that are constant with temperature then the equilibrium constant K obeys ln K a T b c ln T displaystyle ln K frac a T b c ln T for some constants a b and c When c equal to DCp R is not zero we can find the value or values of T where K equals a given value as follows where we use L for ln T a b ln K T c T ln T b ln K e L c L e L a c b ln K c L e L a c e b ln K c L b ln K c e L b ln K c L W a c e b ln K c ln K b c T exp W a c e b ln K c ln K b c displaystyle begin aligned a amp b ln K T cT ln T amp b ln K e L cLe L 5pt frac a c amp left frac b ln K c L right e L 5pt frac a c e frac b ln K c amp left L frac b ln K c right e L frac b ln K c 5pt L amp W left frac a c e frac b ln K c right frac ln K b c 5pt T amp exp left W left frac a c e frac b ln K c right frac ln K b c right end aligned If a and c have the same sign there will be either two solutions or none or one if the argument of W is exactly 1 e The upper solution may not be relevant If they have opposite signs there will be one solution Phase separation of polymer mixtures Edit 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 W functions 39 Wien s displacement law in a D dimensional universe Edit Wien s displacement law is expressed as n max T a c o n s t displaystyle nu max T alpha mathrm const With x h n max k B T displaystyle x h nu max k mathrm B T and d r T x d x 0 displaystyle d rho T left x right dx 0 where r T displaystyle rho T is the spectral energy energy density one finds e x 1 x D displaystyle e x 1 frac x D The solution x D W D e D displaystyle x D W left De D right shows that the spectral energy density is dependent on the dimensionality of the universe 40 AdS CFT correspondence Edit 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 W function 41 42 Epidemiology Edit In the t limit of the SIR model the proportion of susceptible and recovered individuals has a solution in terms of the Lambert W function 43 Determination of the time of flight of a projectile Edit 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 W function Electromagnetic surface wave propagation Edit 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 u ln u v where u and v clump together the geometrical and physical factors of the problem which is solved by the Lambert W function The first solution to this problem due to Sommerfeld circa 1898 already contained an iterative method to determine the value of the Lambert W function 44 Orthogonal trajectories of real ellipsesThe family of ellipses x 2 1 e 2 y 2 e 2 displaystyle x 2 1 varepsilon 2 y 2 varepsilon 2 centered at 0 0 displaystyle 0 0 is parameterized by eccentricity e displaystyle varepsilon The orthogonal trajectories of this family are given by the differential equation 1 y y d y 1 x x d x displaystyle left frac 1 y y right dy left frac 1 x x right dx whose general solution is the family y 2 displaystyle y 2 W 0 x 2 exp 2 C x 2 displaystyle W 0 x 2 exp 2C x 2 Generalizations EditThe standard Lambert W function expresses exact solutions to transcendental algebraic equations in x of the form e c x a 0 x r displaystyle e cx a 0 x r 1 where a0 c and r are real constants The solution isx r 1 c W c e c r a 0 displaystyle x r frac 1 c W left frac c e cr a 0 right Generalizations of the Lambert W function 45 46 47 include An application to general relativity and quantum mechanics quantum gravity in lower dimensions in fact a link unknown prior to 2007 48 between these two areas where the right hand side of 1 is replaced by a quadratic polynomial in x e c x a 0 x r 1 x r 2 displaystyle e cx a 0 left x r 1 right left x r 2 right 2 where r1 and r2 are real distinct constants the roots of the quadratic polynomial Here the solution is a function which has a single argument x but the terms like ri and a0 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 Equation 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 49 Here the right hand side of 1 is replaced by a ratio of infinite order polynomials in x e c x a 0 i 1 x r i i 1 x s i displaystyle e cx a 0 frac displaystyle prod i 1 infty x r i displaystyle prod i 1 infty x s i 3 where ri and si are distinct real constants and x is a function of the eigenenergy and the internuclear distance R Equation 3 with its specialized cases expressed in 1 and 2 is related to a large class of delay differential equations G H Hardy s notion of a false derivative provides exact multiple roots to special cases of 3 50 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 51 Plots EditPlots of the Lambert W function on the complex plane z Re W0 x iy z Im W0 x iy z W0 x iy Superimposition of the previous three plotsNumerical evaluation EditThe W function may be approximated using Newton s method with successive approximations to w W z so z wew being w j 1 w j w j e w j z e w j w j e w j displaystyle w j 1 w j frac w j e w j z e w j w j e w j The W function may also be approximated using Halley s method w j 1 w j w j e w j z e w j w j 1 w j 2 w j e w j z 2 w j 2 displaystyle w j 1 w j frac w j e w j z e w j left w j 1 right dfrac left w j 2 right left w j e w j z right 2w j 2 given in Corless et al 3 to compute W For real x 1 e displaystyle x geq 1 e it could be approximated by the quadratic rate recursive formula of R Iacono and J P Boyd 52 w n 1 x w n x 1 w n x 1 log x w n x displaystyle w n 1 x frac w n x 1 w n x left 1 log left frac x w n x right right Lajos Loczi proves that by choosing appropriate w 0 x displaystyle w 0 x if x e displaystyle x in e infty w 0 x log x log log x displaystyle w 0 x log x log log x if x 0 e displaystyle x in 0 e w 0 x x e displaystyle w 0 x x e if x 1 e 0 displaystyle x in 1 e 0 for the principal branch W 0 displaystyle W 0 w 0 x e x 1 e x 1 e x log 1 1 e x displaystyle w 0 x frac ex 1 ex sqrt 1 ex log 1 sqrt 1 ex for the branch W 1 displaystyle W 1 w 0 x 1 2 1 e x displaystyle w 0 x 1 sqrt 2 1 ex for x 1 e 1 4 displaystyle x in 1 e 1 4 w 0 x log x log log x displaystyle w 0 x log x log log x for x 1, wikipedia, wiki, book, books, library,

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