Lambert W function

Terminology

The Lambert-W function is named after Johann Heinrich Lambert. The main branch W₀ is denoted by Wp in the Digital Library of Mathematical Functions and the branch W₋₁ is denoted by 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.

History

Lambert first considered the related Lambert's Transcendental Equation in 1758, which led to a paper by Leonhard Euler in 1783 that discussed the special case of we^w.

The Lambert W function was "re-discovered" every decade or so in specialized applications. In 1993, when it was reported that the Lambert W function provides an exact solution to the quantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem in physics—Corless and developers of the Maple Computer algebra system made a library search, and found that this function was ubiquitous in nature.

Derivative

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 .

(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 } .

Using the identity e W ( z ) = z / W ( z ) , we get the following equivalent formula which holds for all z ≠ − 1 / e :

d W d z = 1 z + e W ( z ) .

Antiderivative

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

∫ W ( x ) d x = x W ( x ) − x + e W ( x ) + C = x ( W ( x ) − 1 + 1 W ( x ) ) + C .

(The last equation is more common in the literature but does not hold at x = 0 .)

One consequence of which (using the fact that W ( e ) = 1 ) is the identity:

∫ 0 e W ( x ) d x = e − 1

Asymptotic expansions

The Taylor series of W 0 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 − ⋯

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 + ⋯ W 0 ( x ) = L 1 − L 2 + ∑ ℓ = 0 ∞ ∑ m = 1 ∞ ( − 1 ) ℓ [ ℓ + m ℓ + 1 ] m ! L 1 − ℓ − m L 2 m

where L 1 = ln ⁡ ( x ) , L 2 = ln ⁡ ( ln ⁡ ( x ) ) and [ ℓ + m ℓ + 1 ] is a non-negative Stirling number of the first kind. Keeping only the first two terms of the expansion,

W 0 ( x ) = ln ⁡ ( x ) − ln ⁡ ( ln ⁡ ( x ) ) + 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 L 1 = ln ⁡ ( − x ) and L 2 = ln ⁡ ( − ln ⁡ ( − x ) ) .

In it is shown 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 ) .

In it was proven that branch W − 1 can be bounded as follows:

− 1 − 2 u − u < W − 1 ( − e − u − 1 ) < − 1 − 2 u − 2 3 u

for u > 0 .

Integer and complex powers

Integer powers of W 0 also admit simple Taylor (or Laurent) series expansions at 0

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 − ⋯

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 ,

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 W 0 ( x ) / x

( W 0 ( x ) x ) r = exp ⁡ ( − r W 0 ( x ) ) = ∑ n = 0 ∞ r ( n + r ) n − 1 n !   ( − x ) n ,

which holds for any r ∈ C and | x | < e − 1 .

Identities

A few identities follow from definition:

W ( x ⋅ e x ) = x  for  x ≥ 0 W 0 ( x ⋅ e x ) = x  for  x ≥ − 1 W − 1 ( x ⋅ e x ) = x  for  x ≤ − 1

Note that, since f(x) = x⋅e^x is not injective, not always W(f(x)) = x. For fixed x < 0 and x ≠ -1 the equation x⋅e^x = y⋅e^y has two solutions in y, one of which is of course y = x. Then, for i = 0 and x < −1 as well as for i = −1 and x ∈ (−1, 0), W_i(x⋅e^x) is the other solution of the equation x⋅e^x = y⋅e^y.

W ( x ) ⋅ e W ( x ) = x e W ( x ) = x W ( x ) e − W ( x ) = W ( x ) x e n ⋅ W ( x ) = ( x W ( x ) ) n ln ⁡ W ( x ) = ln ⁡ ( x ) − W ( x )  for  x > 0 W ( x ) = ln ⁡ ( x W ( x ) )  for  x ≥ − 1 e W ( n x n W ( x ) n − 1 ) = n ⋅ W ( x )  for  n > 0 ,  x > 0

(which can be extended to other n and x if the right branch is chosen)

W ( x ) + W ( y ) = W ( x y ( 1 W ( x ) + 1 W ( y ) ) )  for  x , y > 0

From inverting f(ln(x)):

W ( x ⋅ ln ⁡ x ) = ln ⁡ x  for  x > 0 W ( x ⋅ ln ⁡ x ) = W ( x ) + ln ⁡ W ( x )  for  x > 0

With Euler's iterated exponential h(x):

h ( x ) = e − W ( − ln ⁡ ( x ) ) = W ( − ln ⁡ ( x ) ) − ln ⁡ ( x )  for  x ≠ 1

Special values

For any non-zero algebraic number x, W(x) is a transcendental number. Indeed, if W(x) is zero then x must be zero as well, and if W(x) is non-zero and algebraic, then by the Lindemann–Weierstrass theorem, e^W(x) must be transcendental, implying that x=W(x)e^W(x) must also be transcendental.

W ( − π 2 ) = π 2 i W ( − ln ⁡ a a ) = − ln ⁡ a ( 1 e ≤ a ≤ e ) W ( − 1 e ) = − 1 W ( 0 ) = 0 W ( 1 ) = Ω = 1 ∫ − ∞ + ∞ d t ( e t − t ) 2 + π 2 − 1 ≈ 0.56714329 … (the Omega constant) W ( 1 ) = e − W ( 1 ) = ln ⁡ ( 1 W ( 1 ) ) = − ln ⁡ W ( 1 ) W ( e ) = 1 W ( − 1 ) ≈ − 0.31813 − 1.33723 i W ′ ( 0 ) = 1

Definite integrals

There are several useful definite integral formulas involving the W function, including the following:

∫ 0 π W ( 2 cot 2 ⁡ ( x ) ) sec 2 ⁡ ( x ) d x = 4 π ∫ 0 ∞ W ( x ) x x d x = 2 2 π ∫ 0 ∞ W ( 1 x 2 ) d x = 2 π

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

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 ⋅ Γ ( 3 2 ) + 2 ⋅ Γ ( 1 2 ) = 2 2 ( 1 2 π ) + 2 ( π ) = 2 2 π

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

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:

W ( z ) = z 2 π ∫ − π π ( 1 − ν cot ⁡ ν ) 2 + ν 2 z + ν csc ⁡ ν e − ν cot ⁡ ν d ν = z π ∫ 0 π ( 1 − ν cot ⁡ ν ) 2 + ν 2 z + ν csc ⁡ ν e − ν cot ⁡ ν d ν

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

Indefinite integrals

∫ W ( x ) x d x = 1 2 ( 1 + W ( x ) ) 2 + C

∫ W ( A exp ⁡ ( B x ) ) d x = 1 2 B ( 1 + W ( A exp ⁡ ( B x ) ) ) 2 + C

Applications

Many equations involving exponentials can be solved using the W function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like Y = Xe^X at which point the W function provides the value of the variable in X.

In other words :

Y = X e X ⟺ X = W ( Y )

Example 1

2 t = 5 t 1 = 5 t 2 t 1 = 5 t e − t ln ⁡ 2 1 5 = t e − t ln ⁡ 2 − ln ⁡ 2 5 = ( − t ln ⁡ 2 ) e ( − t ln ⁡ 2 ) W ( − ln ⁡ 2 5 ) = − t ln ⁡ 2 t = − W ( − ln ⁡ 2 5 ) ln ⁡ 2

More generally, the equation

  p x = a x + b

where

p > 0  and  a ≠ 0

can be transformed via the substitution

− t = x + b a

into

t p t = R = − 1 a p − b a

giving

t = W ( R ln ⁡ p ) ln ⁡ p

which yields the final solution

x = − W ( − ln ⁡ p a p − b a ) ln ⁡ p − b a

Example 2

x x = z ⇒ x ln ⁡ x = ln ⁡ z ⇒ e ln ⁡ x ⋅ ln ⁡ x = ln ⁡ z ⇒ ln ⁡ x = W ( ln ⁡ z ) ⇒ x = e W ( ln ⁡ z ) ,

or, equivalently,

x = ln ⁡ z W ( ln ⁡ z ) ,

since

ln ⁡ z = W ( ln ⁡ z ) e W ( ln ⁡ z )

by definition.

Example 3

x n = e − a x   , ⇒ x n e a x = 1   ,

taking the n-th root

⇒ x e a x n = 1   , ⇒ a x n e a x n = a n

let : y = a x n then

y e y = a n   , ⇒ y = W ( a n ) ⇒ x = n a W ( a n ) ,

Example 4

Whenever the complex infinite exponential tetration

z z z ⋅ ⋅ ⋅

converges, the Lambert W function provides the actual limit value as

c = W ( − ln ⁡ ( z ) ) − ln ⁡ ( z )

where ln(z) denotes the principal branch of the complex log function. This can be shown by observing that

z c = c

if c exists, so

z = c 1 / c ⇒ z − 1 = c − 1 / c ⇒ 1 z = ( 1 c ) 1 / c ⇒ − ln ⁡ ( z ) = ( 1 c ) ln ⁡ ( 1 c ) ⇒ − ln ⁡ ( z ) = e ln ⁡ ( 1 c ) ln ⁡ ( 1 c ) ⇒ ln ⁡ ( 1 c ) = W ( − ln ⁡ ( z ) ) ⇒ 1 c = e W ( − ln ⁡ ( z ) ) ⇒ 1 c = − ln ⁡ ( z ) W ( − ln ⁡ ( z ) ) ⇒ c = W ( − ln ⁡ ( z ) ) − ln ⁡ ( z )

which is the result which was to be found.

Example 5

Solutions for

x log b ⁡ ( x ) = a

have the form

x = e W ( a ln ⁡ b ) .

Example 6

The solution for the current in a series diode/resistor circuit can also be written in terms of the Lambert W. See diode modeling.

Example 7

The delay differential equation

y ˙ ( t ) = a y ( t − 1 )

has characteristic equation λ = a e − λ , leading to λ = W k ( a ) and y ( t ) = e W k ( a ) t , where k is the branch index. If a ≥ e − 1 , only W 0 ( a ) need be considered.

Example 8

The Lambert-W function has been recently (2013) shown to be the optimal solution for the required magnetic field of a Zeeman slower.

Example 9

Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in the laboratory experiments can be described by using the Lambert–Euler omega function as follows:

H ( x ) = 1 + W [ ( H ( 0 ) − 1 ) exp ⁡ ( ( H ( 0 ) − 1 ) − x L ) ] ,

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.

Example 10

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.

Example 11

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.

Example 12

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.

Example 13

The Lambert-W function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneus tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the -1 branch applies if the displacement is unstable with the heavier fluid running underneath the ligther fluid.

Example 14

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

Y = X 1 − e X

can be solved by means of the two real branches W 0 and W − 1 :

X ( Y ) = W − 1 ( Y e Y ) − W 0 ( Y e Y ) = Y − W 0 ( Y e Y ) for  Y < − 1. X ( Y ) = W 0 ( Y e Y ) − W − 1 ( Y e Y ) = Y − W − 1 ( Y e Y ) for  − 1 < Y < 0.

This application shows in evidence that the branch difference of the W function can be employed in order to solve other trascendental equations.

See : D. J. Jeffrey and J. E. Jankowski, "Branch differences and Lambert W"

Example 15

The centroid of a set of histograms defined with respect to the symmetrized Kullback-Leibler divergence (also called the Jeffreys divergence) is in closed form using the Lambert function.

See : F. Nielsen, "Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms"

Example 16

The Lambert-W function appears in a quantum-mechanical potential (see The Lambert-W step-potential) which affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potential – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as

V = V 0 1 + W ( e − x / σ ) .

A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to z = W ( e − x / σ ) .

See : A.M. Ishkhanyan, "The Lambert W-barrier – an exactly solvable confluent hypergeometric potential"

Generalizations

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

e − c x = a o ( x − r )     ( 1 )

where a₀, c and r are real constants. The solution is x = r + 1 c W ( c e − c r a o ) . Generalizations of the Lambert W function include:

An application to general relativity and quantum mechanics (quantum gravity) in lower dimensions, in fact a previously unknown link (unknown prior to) between these two areas, where the right-hand-side of (1) is now a quadratic polynomial in x:

e − c x = a o ( x − r 1 ) ( x − r 2 )     ( 2 ) and where r₁ and r₂ are real distinct constants, the roots of the quadratic polynomial. Here, the solution is a function has a single argument x but the terms like r_i and a_o are parameters of that function. In this respect, the generalization resembles the hypergeometric function and the Meijer G-function but it belongs to a different class of functions. When r₁ = r₂, both sides of (2) can be factored and reduced to (1) and thus the solution reduces to that of the standard W function. Eq. (2) expresses the equation governing the dilaton field, from which is derived the metric of the R=T or lineal two-body gravity problem in 1+1 dimensions (one spatial dimension and one time dimension) for the case of unequal (rest) masses, as well as, the eigenenergies of the quantum-mechanical double-well Dirac delta function model for unequal charges in one dimension.

Analytical solutions of the eigenenergies of a special case of the quantum mechanical three-body problem, namely the (three-dimensional) hydrogen molecule-ion. Here the right-hand-side of (1) (or (2)) is now a ratio of infinite order polynomials in x:

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

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.

Plots

Plots of the Lambert ''W'' function on the complex plane

Numerical evaluation

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

w j + 1 = w j − 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

given in Corless et al. to compute W.

Software

The Lambert-W function is implemented as LambertW in Maple, lambertw in GP (and glambertW in PARI), lambertw in MATLAB, also lambertw in octave with the 'specfun' package, as lambert_w in Maxima, as ProductLog (with a silent alias LambertW) in Mathematica, as lambertw in Python scipy's special function package, as LambertW in Perl's ntheory module, and as gsl_sf_lambert_W0 and gsl_sf_lambert_Wm1 functions in special functions section of the GNU Scientific Library – GSL.