Lambert W function 1

Lambert W function

The graph of W(x) for W > −4 and x < 6. The upper branch with W ≥ −1 is thefunction W0 (principal branch), the lower branch with W ≤ −1 is the function

W−1.

In mathematics, the Lambert W function,also called the Omega function or productlogarithm, is a set of functions, namely thebranches of the inverse relation of the functionf(w) = wew where ew is the exponentialfunction and w is any complex number. Inother words, the defining equation for W(z) is

for any complex number z.Since the function ƒ is not injective, the relationW is multivalued (except at 0). If we restrictattention to real-valued W then the relation isdefined only for x ≥ −1/e, and is double-valuedon (−1/e, 0); the additional constraint W ≥ −1defines a single-valued function W0(x). Wehave W0(0) = 0 and W0(−1/e) = −1.Meanwhile, the lower branch has W ≤ −1 andis denoted W−1(x). It decreases from W−1(−1/e) = −1 to W−1(0−) = −∞.

The Lambert W relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, forinstance in the enumeration of trees. It can be used to solve various equations involving exponentials (e.g. themaxima of the Planck, Bose-Einstein, and Fermi-Dirac distributions) and also occurs in the solution of delaydifferential equations, such as y'(t) = a y(t − 1).

Lambert W function 2

Main branch of the Lambert W function in the complex plane. Note the branchcut along the negative real axis, ending at −1/e. In this picture, the hue of a

point z is determined by the argument of W(z) and the brightness by theabsolute value of W(z).

Terminology

The Lambert W-function is named after JohannHeinrich Lambert. The main branch W0 isdenoted by Wp in the Digital Library ofMathematical Functions and the branch W−1 isdenoted by Wm there.

The notation convention chosen here (with W0and W−1) follows the canonical reference onthe Lambert-W function by Corless, Gonnet,Hare, Jeffrey and Knuth.[1]

History

Lambert first considered the related Lambert'sTranscendental Equation in 1758,[2] which ledto a paper by Leonhard Euler in 1783[3] thatdiscussed the special case of wew. However theinverse of wew was first described by Pólyaand Szegő in 1925.[4] The Lambert W functionwas "re-discovered" every decade or so inspecialized applications but its full importancewas not realized until the 1990s. When it was reported that the Lambert W function provides an exact solution to thequantum-mechanical double-well Dirac delta function model for equal charges—a fundamental problem inphysics—Corless and developers of the Maple Computer algebra system made a library search to find that thisfunction was in fact ubiquitous to nature.[5]

Calculus

DerivativeBy 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:

Furthermore we have

Lambert W function 3

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

Asymptotic expansionsThe Taylor series of 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 beextended 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.

An asymptotic expansion for the other real branch, , defined in the interval (−∞, −1/e], is

where and and is a non-negative Stirling numbers of the first

kind.

Integer and complex powers

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

More generally, for , 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 Taylorexpansion of powers of

which holds for any and .

Lambert W function 4

Special valuesFor any non-zero algebraic number x, W(x) is a transcendental number. We can show this by contradiction: If W(x)were non-zero and algebraic (note that if x is non-zero then W(x) must be non-zero as well), then by theLindemann–Weierstrass theorem, eW(x) must be transcendental, implying that x=W(x)eW(x) must also betranscendental, contradicting the condition that x is algebraic.

(the Omega constant)

Other formulas

ApplicationsMany equations involving exponentials can be solved using the W function. The general strategy is to move allinstances of the unknown to one side of the equation and make it look like Y = XeX at which point the W functionprovides the value of the variable in X.In other words :

Lambert W function 5

Examples

Example 1

More generally, the equation

where

can be transformed via the substitution

into

giving

which yields the final solution

Example 2

or, equivalently,

since

Lambert W function 6

by definition.

Example 3

Whenever the complex infinite exponential tetration

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

where ln(z) denotes the principal branch of the complex log function.

Example 4

Solutions for

have the form

[6]

Example 5

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

Example 6

The delay differential equation

has characteristic equation , leading to and , where is the branchindex. If , only need be considered.

Example 7

Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in the laboratoryexperiments can be described by using the Lambert–Euler omega function as follows:[7]

where H(x) is the debris flow height, x is the channel downstream position, L is the unified model parameterconsisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressuregradient.

Lambert W function 7

Example 8

The Lambert W function was employed in the field of Neuroimaging for linking cerebral blood flow and oxygenconsumption changes within a brain voxel, to the corresponding Blood Oxygenation Level Dependent (BOLD)signal[8].

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

where a0, c and r are real constants. The solution is . Generalizations of the Lambert W

function[9] include:• An application to general relativity and quantum mechanics (quantum gravity) in lower dimensions, in fact a

previously unknown link (unknown prior to the 2007 paper by Farrugia, Mann and Scott[10]) between these twoareas, where the right-hand-side of (1) is now a quadratic polynomial in x:

and where r1 and r2 are real distinct constants, the roots of the quadratic polynomial. Here, the solution is afunction has a single argument x but the terms like ri and ao are parameters of that function. In this respect, thegeneralization resembles the hypergeometric function and the Meijer G-function but it belongs to a differentclass of functions. When r1 = r2, both sides of (2) can be factored and reduced to (1) and thus the solutionreduces to that of the standard W function. Eq. (2) expresses the equation governing the dilaton field, fromwhich is derived the metric of the R=T or lineal two-body gravity problem in 1+1 dimensions (one spatialdimension and one time dimension) for the case of unequal (rest) masses, as well as, the eigenenergies of thequantum-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.[11] Here the right-hand-side of (1) (or (2)) is now a ratioof infinite order polynomials in x:

where ri and si 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 differentialequations.

Applications of the Lambert "W" function in fundamental physical problems are not exhausted even for the standardcase expressed in (1) as seen recently in the area of atomic, molecular, and optical physics.[12]

Lambert W function 8

Plots

Plots of the Lambert W function on the complex plane

z = Re(W0(x + i y)) z = Im(W0(x + i y))

Numerical evaluationThe W function may be approximated using Newton's method, with successive approximations to (so

) being

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

given in Corless et al. to compute W.

SoftwareThe LambertW function is implemented as lambertw in MATLAB[13] and as ProductLog in Mathematica[14].

Notes[1] Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. (1996). "On the Lambert W function" (http:/ / www. apmaths. uwo.

ca/ ~rcorless/ frames/ PAPERS/ LambertW/ LambertW. ps). Advances in Computational Mathematics 5: 329–359. doi:10.1007/BF02124750..

[2] Lambert JH, "Observationes variae in mathesin puram", Acta Helveticae physico-mathematico-anatomico-botanico-medica, Band III,128–168, 1758 ( facsimile (http:/ / www. kuttaka. org/ ~JHL/ L1758c. pdf))

[3] Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler,L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921. ( facsimile (http:/ /math. dartmouth. edu/ ~euler/ docs/ originals/ E532. pdf))

[4] Pólya, George; Szegő, Gábor (1998) [1925]. Aufgaben und Lehrsätze der Analysis [Problems and Theorems in Analysis]. Berlin:Springer-Verlag.

[5] Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J. (1993). "Lambert's W function in Maple". The Maple Technical Newsletter(MapleTech) 9: 12–22.

[6] Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J. (1993). "Lambert's W function in Maple". The Maple Technical Newsletter(MapleTech) 9: 12–22.

[7] Pudasaini, S.P. (2011). "Some exact solutions for debris and avalanche flows". Physics of Fluids 23 (4): 043301. doi:10.1063/1.3570532.[8] Sotero, Roberto C.; Iturria-Medina, Yasser (2011). "From Blood oxygenation level dependent (BOLD) signals to brain temperature maps".

Bull Math Biol 73 (11): 2731–47. doi:10.1007/s11538-011-9645-5. PMID 21409512.[9] Scott, T. C.; Mann, R. B.; Martinez Ii, Roberto E. (2006). "General Relativity and Quantum Mechanics: Towards a Generalization of the

Lambert W Function". AAECC (Applicable Algebra in Engineering, Communication and Computing) 17 (1): 41–47. arXiv:math-ph/0607011.doi:10.1007/s00200-006-0196-1.

[10] Farrugia, P. S.; Mann, R. B.; Scott, T. C. (2007). "N-body Gravity and the Schrödinger Equation". Class. Quantum Grav. 24 (18):4647–4659. arXiv:gr-qc/0611144. doi:10.1088/0264-9381/24/18/006.

Lambert W function 9

[11] Scott, T. C.; Aubert-Frécon, M.; Grotendorst, J. (2006). "New Approach for the Electronic Energies of the Hydrogen Molecular Ion". Chem.Phys. 324 (2–3): 323–338. arXiv:physics/0607081. doi:10.1016/j.chemphys.2005.10.031.

[12] Scott, T. C.; Lüchow, A.; Bressanini, D.; Morgan, J. D. III (2007). "The Nodal Surfaces of Helium Atom Eigenfunctions". Phys. Rev. A 75(6): 060101. doi:10.1103/PhysRevA.75.060101.

[13] lambertw - MATLAB (http:/ / www. mathworks. com. au/ help/ toolbox/ symbolic/ lambertw. html)[14] ProductLog at WolframAlpha (http:/ / reference. wolfram. com/ mathematica/ ref/ ProductLog. html)

References• Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.; Knuth, Donald (1996). "On the Lambert W function" (http:/ / www.

apmaths. uwo. ca/ ~djeffrey/ Offprints/ W-adv-cm. pdf). Advances in Computational Mathematics (Berlin, NewYork: Springer-Verlag) 5: 329–359. doi:10.1007/BF02124750. ISSN 1019-7168

• Chapeau-Blondeau, F. and Monir, A: "Evaluation of the Lambert W Function and Application to Generation ofGeneralized Gaussian Noise With Exponent 1/2", IEEE Trans. Signal Processing, 50(9), 2002 (http:/ / www. istia.univ-angers. fr/ ~chapeau/ papers/ lambertw. pdf)

• Francis et al. "Quantitative General Theory for Periodic Breathing" Circulation 102 (18): 2214. (2000). (http:/ /circ. ahajournals. org/ cgi/ reprint/ 102/ 18/ 2214) Use of Lambert function to solve delay-differential dynamics inhuman disease.

• Roy, R.; Olver, F. W. J. (2010), "Lambert W function" (http:/ / dlmf. nist. gov/ 4. 13), in Olver, Frank W. J.;Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge UniversityPress, ISBN 978-0521192255, MR2723248

• Veberic, D., "Having Fun with Lambert W(x) Function" arXiv:1003.1628 (2010). (http:/ / arxiv. org/ abs/ 1003.1628) C++ implementation (http:/ / www. ung. si/ ~darko/ LambertW/ ) using Halley's and Fritsch's iteration.

External links• National Institute of Science and Technology Digital Library - Lambert W (http:/ / dlmf. nist. gov/ 4. 13)• MathWorld - Lambert W-Function (http:/ / mathworld. wolfram. com/ LambertW-Function. html)• Computing the Lambert W function (http:/ / www. whim. org/ nebula/ math/ lambertw. html)• Corless et al. Notes about Lambert W research (http:/ / www. apmaths. uwo. ca/ ~rcorless/ frames/ PAPERS/

LambertW/ )• Extreme Mathematics. (http:/ / ioannis. virtualcomposer2000. com/ math/ ) Monographs on the Lambert W

function, its numerical approximation and generalizations for W-like inverses of transcendental forms withrepeated exponential towers.

• GPL C++ implementation (http:/ / www. ung. si/ ~darko/ LambertW. tar. gz) with Halley's and Fritsch's iteration.

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