The Lambert W Function

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III.17. The Lambert W Function 151 faster than shorter ones, Zabusky and Kruskal simu- quantities, infinitely many higher-order symmetries, lated the collision of two waves in a nonlinear crys- and an infinite number of soliton solutions. tal lattice and observed that each retains its shape and As well as being applicable to shallow-water waves, speed after collision. Interacting solitary waves merely the KdV equation is ubiquitous in applied science. It experience a phase shift, advancing the faster wave and describes, for example, ion-acoustic waves in a plasma, retarding the slower one. In analogy with colliding par- elastic waves in a rod, and internal waves in the atmo- ticles, they coined the word “solitons” to describe these sphere or ocean. The KP equation models, for exam- elastically colliding waves. ple, water waves, acoustic waves, and magnetoelastic To model water waves that are weakly nonlinear, waves in anti-ferromagnetic materials. The nonlinear weakly dispersive, and weakly two-dimensional, with Schrödinger equation describes weakly nonlinear and all three effects being comparable, Kadomtsev and dispersive wave packets in physical systems, e.g., light Petviashvili (KP) derived a two-dimensional version of pulses in optical fibers, surface waves in deep water, (2) in 1970: Langmuir waves in a plasma, and high-frequency vibra- 2 tions in a crystal lattice. Equation (7) with an extra lin- (ut + 6uux + uxxx)x + 3σ uyy = 0, (6) ear term V(x)u to account for the external potential 2 =± where σ 1 and the y-axis is perpendicular to the V(x) also arises in the study of Bose–Einstein conden- direction of propagation of the wave (along the x-axis). sates, where it is referred to as the time-dependent The KdV and KP equations, and the nonlinear Gross–Pitaevskii equation. schrödinger equation [III.26] 2 Further Reading iut + uxx + κ|u| u = 0 (7) (where κ is a constant and u(x, t) is a complex-valued Ablowitz, M. J. 2011. Nonlinear Dispersive Waves: Asymp- function), are famous examples of so-called completely totic Analysis and Solitons. Cambridge: Cambridge Univer- sity Press. integrable nonlinear PDEs. This means that they can Ablowitz, M. J., and P. A. Clarkson. 1991. Solitons, Nonlinear be solved with the inverse scattering transform, a Evolution Equations and Inverse Scattering. Cambridge: nonlinear analogue of the Fourier transform. Cambridge University Press. The inverse scattering transform is not applied to (2) Kasman, A. 2010. Glimpses of Soliton Theory. Providence, directly but to an auxiliary system of linear PDEs, RI: American Mathematical Society. Osborne, A. R. 2010. Nonlinear Ocean Waves and the Inverse + + 1 = ψxx (λ 6 αu)ψ 0, (8) Scattering Transform. Burlington, MA: Academic Press. + 1 + + = ψt 2 αuxψ αuψx 4ψxxx 0, (9) which is called the Lax pair for the KdV equation. Equa- III.17 The Lambert W Function tion (8) is a linear Schrödinger equation for an eigen- Robert M. Corless and David J. Jeffrey function ψ, a constant eigenvalue λ, and a potential (−αu)/6. Equation (9) governs the time evolution of ψ. 1 Definition and Basic Properties The two equations are compatible, i.e., ψxxt = ψtxx, if and only if u(x, t) satisfies (2). For given u(x, 0) For a given complex number z, the equation decaying sufficiently fast as |x|→∞, the inverse scat- wew = z tering transform solves (8) and (9) and finally deter- has a countably infinite number of solutions, which are mines u(x, t). denoted by Wk(z) for integers k. Each choice of k speci- fies a branch of the Lambert W function. By convention, 4 Properties and Applications only the branches k = 0 (called the principal branch) =− Scientists remain intrigued by the rich mathemati- and k 1 are real-valued for any z; the range of every cal structure of completely integrable nonlinear PDEs. other branch excludes the real axis, although the range −∞ − These PDEs can be written as infinite-dimensional bi- of W1(z) includes ( , 1/e] in its closure. Only W0(z) Hamiltonian systems and have additional, remarkable contains positive values in its range (see figure 1). When =− features. For example, they have an associated Lax pair, z 1/e (the only nonzero branch point), there is a =− w = they can be written in Hirota’s bilinear form, they admit double root w 1 of the basic equation we z. Bäcklund transformations, and they have the Painlevé The conventional choice of branches assigns property. They have an infinite number of conserved W0(−1/e) = W−1(−1/e) =−1

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152 III. Equations, Laws, and Functions of Applied Mathematics

W W 1 0 3

2 –e–1 z –1 1 2 3 1 )

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( z

–1 k 0 ( W

Im –1 –2

–2

–3 –3 W–1

–4 –3 –2 –1 3210 Re (Wk (z ) ) Figure 1 Real branches of the Lambert W function. The solid line is the principal branch W0; the dashed line is W−1, Figure 2 Images of circles and rays in the z-plane under the −1 which is the only other branch that takes real values. The maps z → Wk(z). The circle with radius e maps to a curve small filled circle at the branch point corresponds to the that goes through the branch point, as does the ray along one in figure 2. the negative real axis. This graph was produced in Maple x+iy by numerical evaluation of ω(x + iy) = WK(iy)(e ) first for a selection of fixed x and varying y, and then for a 2 and implies that W1(−1/e−iε ) =−1+O(ε) is arbitrar- selection of fixed y and varying x. These two sets produce ily close to −1 because the conventional branch choice orthogonal curves as images of horizontal and vertical lines means that the point −1 is on the border between these in x and y under ω or, equivalently, images of circles with = x = three branches. Each branch is a single-valued com- constant r e and rays with constant θ y under W . plex function, analytic away from the branch point and branch cuts. ln z is the principal branch of the logarithm, having The set of all branches is often referred to, loosely, as −πsine or cosine, the branches of W are not linearly related. However, by Implicit differentiation yields rephrasing things slightly, in terms of the unwinding = −W(z) + number W (z) e /(1 W(z)) z − ln(ez) K(z) := =− 2πi as long as W(z) 1. The derivative can be simplified and the related single-valued function to the rational differential equation = z dW W ω(z) : WK(z)(e ), = dz z(1 + W) which is called the Wright ω function, we do have the somewhat simple relationship between branches that if, in addition, z = 0. Higher derivatives follow natu-

Wk(z) = ω(lnk z), where lnk z denotes ln z + 2πik and rally.

III.17. The Lambert W Function 153

1.2 Integrals is split at the maximum of the integrand (t = n), and each integral is transformed using the substitu- Integrals containing W(x)can often be performed ana- 2 tions t =−nW (−e−1−z /2), where k = 0 is used for = k lytically by the change of variable w W(x), used in =− w t n and k 1 otherwise. The integrands then an inverse fashion: x = we . Thus, 2 simplify to tne−t = nne−ne−nz /2 and the differentials sin W(x)dx = (1 + w)ew sin w dw, dt are obtained as series from the above expansions. Term-by-term integration leads to and integration using usual methods gives n+1 k+1/2 1 1 n 2 1 (1 + w)ew sin w − wew cos w, n! ∼ (2k + 1)a2k+1 Γ(k+ ), 2 2 en n 2 k0 which eventually gives where Γ is the gamma function. x →∞ 2 sin W(x)dx = x + sin W(x) Asymptotic series for z have been known since W(x) de Bruijn’s work in the 1960s. He also proved that − x cos W(x)+ C. the asymptotic series are actually convergent for large More interestingly, there are many definite integrals for enough z. The series begin as follows: Wk(z) ∼ lnk(z)− W(z), including one for the principal branch that is due ln(lnk(z)) + o(ln lnk z). Somewhat surprisingly, these to Poisson and is listed in the famous table of integrals series can be reversed to give a simple (though appar- by D. Bierens de Haan. The following integral, which ently useless) expansion for the logarithm in terms of is of relatively recent construction and which is valid compositions of W : for z not in (−∞, −1/e], can be computed with spectral ln z = W(z)+ W(W(z))+ W(W(W(z)))+··· accuracy by the trapezoidal rule: + W (N)(z) + ln W (N)(z) W(z) 1 π (1 − v cot v)2 + v2 = dv. −v cot v z 2π −π z + v csc ve for a suitably restricted domain in z. The series obtained by omitting the term ln W (N)(z) is not con- 1.3 Series and Generating Functions vergent as N →∞, but for fixed N if we let z →∞ Euler was the first to notice, using a series due to Lam- the approximation improves, although only tediously bert, that what we now call the Lambert W function has slowly. a convergent series expansion around z = 0: 2 Applications (−n)n−1 W(z) = zn. n! n1 Because W is a so-called implicitly elementary function, meaning it is defined as an implicit solution of Euler knew that this series converges for −1/e z an equation containing only elementary functions, it 1/e. The nearest singularity is the branch point z = can be considered an “answer” rather than a question. −1/e. That it solves a simple rational differential equation W can also be expanded in series about the branch means that it occurs in a wide range of mathematical point. The series at the branch point can be expressed models. Out of many applications, we mention just two most cleanly using the tree function T(z) =−W(−z) favorites. rather than W or ω, but keeping with W we have First, a serious application. W occurs in a chemi- − −1−z2/2 =− − n n W0( e ) ( 1) anz , cal kinetics model of how the human eye adapts to n0 darkness after exposure to bright light: a phenomenon −1−z2/2 n W−1(−e ) =− anz , known as bleaching. The model differential equation is n0 K O (t) d = m p where the an are given by a0 = a1 = 1 and Op(t) , dt τ(Km + Op(t)) n−1 1 and its solution in terms of W is an = an−1 − kakan+1−k . (n + 1)a1 B k=2 B/Km−t/τ Op(t) = KmW e , These give an interesting variation on stirling’s for- Km mula [IV.7 §3] for the asymptotics of n!. Euler’s integral where the constant B is the initial value of Op(0), ∞ that is, the amount of initial bleaching. The constants = n −t n! t e dt Km and τ are determined by experiment. The solution 0

154 III. Equations, Laws, and Functions of Applied Mathematics in terms of W enables more convenient analysis by of branch differences of W : a solution of allowing the use of known properties. z + v csc ve−v cot v = 0 The second application we mention is nearly frivo- is v = (W (z) − W (z))/(2i) for some pair of integers lous. W can be used to explore solutions of the so-called k - k and -; moreover, every such pair generates a solu- astrologer’s equation, y(t)˙ = ay(t + 1). In this equation. This bi-infinite family of solutions has accumu- tion, the rate of change of y is supposed to be propor- lation points of zeros near odd multiples of π, which tional to the value of y one time unit into the future. in turn implies that the denominator in the above def- Dependence on past times instead leads to delay dif- inite integral for W (z)/z has essential singularities at ferential equations [I.2 §12], which of course are of v =±π. This example underlines the importance of serious interest in applications, and again W is use- the fact that the branches of W are not trivially related. ful there in much the same way as for this frivolous Another equation of popular interest occurs in the problem. analysis of the limit of the recurrence relation Frivolity can be educational, however. Notice first = an that, if eλt satisfies the equation, then λ = aeλ, and an+1 z =− − therefore λ Wk( a). For the astrologer’s equation, starting with, say, a0 = 1. This sequence has a1 = = z = zz any function y(t) that can be expressed as a finite lin- z, a2 z , a3 z , and so on. If this limit − − = Wk( a)t ear combination y(t) k∈M cke for 0 t converges, it does so to a solution of the equation 1 and some finite set M of integers then solves the a = za. By inspection, the limit that is of interest astrologer’s equation for all time. Thus, perfect know- is a =−W(− ln z)/ ln z. Somewhat surprisingly, this ledge of y(t) on the time interval 0 t 1 is sufficient recurrence relation—which defines the so-called tower to predict y(t) for all time. However, if the knowledge of exponentials—diverges for small enough z, even of y(t) is imperfect, even by an infinitesimal amount if z is real. Specifically, the recurrence converges for − − − (omitting a single term εe WK ( a)t , say, where K is some e e z e1/e if z is real and diverges if z< −e large integer), then since the real parts of −WK (−a) go e = 0.0659880 .... This fact was known to Euler. The to infinity as K →∞by the first two terms of the log- detailed convergence properties for complex z were arithmic series for Wk given above, the “true” value of settled only relatively recently. Describing the regions y(t) can depart arbitrarily rapidly from the prediction. in the complex plane where the recurrence relation con- This seems completely in accord with our intuitions verges to an n-cycle is made possible by a transforma- about horoscopes. tion that is itself related to W :ifζ =−W(− ln z), then Returning to serious applications, we note that the the iteration converges if |ζ| < 1, and also if ζ = eiθ tree function T(z) has huge combinatorial significance for θ equal to some rational multiple of π, say mπ/k. for all kinds of enumeration. Many instances can be Regions where the iteration converges to a k-cycle may found in Knuth’s selected papers, for example. Addi- touch the unit circle at those points. tionally, a key reference to the tree function is a note by Borel in Comptes Rendus de l’Académie des Sciences 4 Retrospective (volume 214, 1942; reprinted in his Œuvres). The gen- The Lambert W function crept into the mathematics lit- erating function for probabilities of the time between erature unobtrusively, and it now seems natural there. periods when a queue is empty, given Poisson arrivals There is even a matrix version of it, although the solu- and service time σ ,isT(σe−σ z)/σ . tion of the matrix equation SeS = A is not always W (A). Hindsight can, as it so often does, identify the pres- 3 Solution of Equations ence of W in writings by Euler, Poisson, and Wright Several equations containing algebraic quantities to- and in many applications. Its implementation in Maple in the early 1980s was a key step in its eventual gether with logarithms or exponentials can be manip- popularity. ulated into either the form y + ln y = z or wew = z, Indeed, its recognition and naming supports Alfred and hence solved in terms of the Lambert W function. North Whitehead’s opinion that: However, it appears that not every exponential polynomial equation—or even most of them—can be solved in By relieving the brain of all unnecessary work, a good this way. We point out one equation, here, that starts notation sets it free to concentrate on more advanced with a nested exponential and can be solved in terms problems.

III.18. Laplace’s Equation 155

Further Reading A short list of solutions, V(x,y), for (2) follows:

Corless, R. M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and • there are polynomial solutions, such as 1, x, y, D. E. Knuth. 1996. On the Lambert W function. Advances xy, and x2 − y2; in Computational Mathematics 5(1):329–59. • there are solutions such as eαx cos αy and eαx × Knuth, D. E. 2000. Selected Papers on the Analysis of Algo- rithms. Palo Alto, CA: Stanford University Press. sin αy, where α is an arbitrary parameter; and α α . 2003. Selected Papers on Discrete Mathematics. Stan- • log r , θ, r cos αθ, and r sin αθ are solutions, ford, CA: CSLI Publications. where x = r cos θ and y = r sin θ define plane Lamb, T., and E. Pugh. 2004. Dark adaptation and the polar coordinates, r and θ. retinoid cycle of vision. Progress in Retinal and Eye Research 23(3):307–80. Further solutions can be found by differentiating or Olver, F. W. J., D. W. Lozier, R. F. Boisvert, and C. W. integrating any solution with respect to x or y; for Clark, eds. 2010. NIST Handbook of Mathematical Func- example, (∂/∂x) log r = x/r2 is harmonic. One can also tions. Cambridge: Cambridge University Press. (Electronic differentiate or integrate with respect to any parameter; version available at http://dlmf.nist.gov.) for example, α2 g(α) eαx sin αy dα III.18 Laplace’s Equation α1 P. A. Martin is harmonic, where g(α) is an arbitrary (integrable) function of the parameter α and the integration could

In 1789, Pierre-Simon Laplace (1749–1827) wrote down be over the real interval α1 <α<α2 or along a contour an equation, in the complex α-plane. ∂2V ∂2V ∂2V A list of solutions, V(x,y,z), for (1) follows: + + = 0, (1) ∂x2 ∂y2 ∂z2 • any solution of (2) also solves (1); that now bears his name. Today, it is arguably the • there are polynomial solutions, such as xyz and most important partial differential equation (PDE) in x2 + y2 − 2z2; mathematics. • there are solutions such as eγz cos αx cos βy, The left-hand side of (1) defines the Laplacian of V , where γ2 = α2 + β2 and α and β are arbitrary denoted by ∇2V or ΔV : parameters; ∇2 = =∇·∇ = αz V ΔV V div grad V. • e Jn(αr ) cos nθ is a solution, where x = r cos θ, Laplace’s equation, ∇2V = 0, is classified as a linear y = r sin θ, and Jn is a Bessel function; and −1 2 2 2 −1/2 homogeneous second-order elliptic PDE for V(x,y,z). • R is a solution, where R = (x + y + z ) is The inhomogeneous version, ∇2V = f , where f is a spherical polar coordinate. a given function, is known as Poisson’s equation. The Again, further solutions can be obtained by differenti- fact that there are three independent variables (x, y, ating or integrating any solution with respect to x, y, and z) in (1) can be indicated by calling it the three- z, or any parameter. For example, if i, j, and k are any dimensional Laplace equation. The two-dimensional nonnegative integers, version, + + ∂2V ∂2V ∂i j k 1 ∇2V ≡ + = 0, (2) V(x,y,z) = ∂x2 ∂y2 ∂xi∂yj ∂zk R also has important applications; it is a PDE for V(x,y). is harmonic; it is known as a Maxwell multipole. There is a natural generalization to n independent As Laplace’s equation is linear and homogeneous, variables. Usually, the number of terms in ∇2V is more solutions can be constructed by superposition; 2 determined by the context. if V1 and V2 satisfy ∇ V = 0, then so does AV1 + BV2, where A and B are arbitrary constants. 1 Harmonic Functions 2 Boundary-Value Problems Solutions of Laplace’s equation are known as harmonic functions. It is easy to see (one of Laplace’s Although Laplace’s equation has many solutions, it is favorite phrases) that there are infinitely many different usual to seek solutions that also satisfy boundary con- harmonic functions. ditions. A basic problem is to solve ∇2V = 0 inside a

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