{"history":[],"article":{"id":"9f02c079-316a-4c0a-abe3-02c398cd1ac8","title":"Lambert W function","slug":"lambert-w-function","content":":::summary\n:::\n\n# \n[[File:The product logarithm Lambert W function plotted in the complex plane from -2-2i to 2+2i.svg|upright=1.3|alt=The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i|thumb|The product logarithm Lambert W function plotted in the complex plane from\n−2 − 2i\nto\n2 + 2i\n]] [[File:Mplwp lambert W branches.svg|thumb|upright=1.3|The graph of\ny = W(x)\nfor real\nx < 6\nand\ny > −4\n. The upper branch (blue) with\ny ≥ −1\nis the graph of the function\nW0\n(principal branch), the lower branch (magenta) with\ny ≤ −1\nis the graph of the function\nW−1\n. The minimum value of\nx\nis at\n]]\nIn [mathematics](https://bliptext.com/articles/mathematics), the **Lambert W function**, also called the **omega function** or **product logarithm**, is a [multivalued function](https://bliptext.com/articles/multivalued-function), namely the [branches](https://bliptext.com/articles/branch-point) of the [converse relation](https://bliptext.com/articles/converse-relation) of the function\nf(w) = wew\n, where w is any [complex number](https://bliptext.com/articles/complex-number) and\new\nis the [exponential function](https://bliptext.com/articles/exponential-function).\nThe function is named after [Johann Lambert](https://bliptext.com/articles/johann-heinrich-lambert), who considered a related problem in 1758.\nBuilding on Lambert's work, [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) described the W function per se in 1783.\nFor each integer\nk\nthere is one branch, denoted by\nWk(z)\n, which is a complex-valued function of one complex argument.\nW0\nis known as the [principal branch](https://bliptext.com/articles/principal-branch).\nThese functions have the following property: if\nz\nand\nw\nare any complex numbers, then\nholds if and only if\nWhen dealing with real numbers only, the two branches\nW0\nand\nW−1\nsuffice: for real numbers\nx\nand\ny\nthe equation\ncan be solved for\ny\nonly if\nx ≥ −1⁄e\ny = W0(x)\nif\nx ≥ 0\nand the two values\ny = W0(x)\nand\ny = W−1(x)\nif\n−1⁄e ≤ x < 0\n.\nThe Lambert W function's branches cannot be expressed in terms of [elementary functions](https://bliptext.com/articles/elementary-function).\nIt is useful in [combinatorics](https://bliptext.com/articles/combinatorics), for instance, in the enumeration of [trees](https://bliptext.com/articles/tree-graph).\nIt can be used to solve various equations involving exponentials (e.g. the maxima of the [Planck](https://bliptext.com/articles/planck-s-law), [Bose–Einstein](https://bliptext.com/articles/bose-einstein-distribution), and [Fermi–Dirac](https://bliptext.com/articles/fermi-dirac-distribution) distributions) and also occurs in the solution of [delay differential equations](https://bliptext.com/articles/delay-differential-equation), such as\ny′(t) = a y(t − 1)\n. In [biochemistry](https://bliptext.com/articles/biochemistry), and in particular [enzyme kinetics](https://bliptext.com/articles/enzyme-kinetics), an opened-form solution for the time-course kinetics analysis of [Michaelis–Menten kinetics](https://bliptext.com/articles/michaelis-menten-kinetics) is described in terms of the Lambert W function.\nCplot Lambert W.png. Note the [branch cut](https://bliptext.com/articles/branch-cut) along the negative real axis, ending at\n−1⁄e\n.]] [[File:Lambert-modulus-HSV.png|thumb|upright=1.3|The modulus of the principal branch of the Lambert W function, colored according to\narg W(z)\n]]\n\n# Terminology\nThe notation convention chosen here (with\nW0\nand\nW−1\n) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and [Knuth](https://bliptext.com/articles/donald-knuth).\nThe name \"product logarithm\" can be understood as this: Since the [inverse function](https://bliptext.com/articles/inverse-function) of\nf(w) = ew\nis called the [logarithm](https://bliptext.com/articles/logarithm), it makes sense to call the inverse \"function\" of the [product](https://bliptext.com/articles/product-mathematics)\nwew\nas \"product logarithm\".\n(Technical note: like the [complex logarithm](https://bliptext.com/articles/complex-logarithm), it is multivalued and thus W is described as the [converse relation](https://bliptext.com/articles/converse-relation) rather than inverse function.) It is related to the [omega constant](https://bliptext.com/articles/omega-constant), which is equal to\nW0(1)\n.\n\n# History\nLambert first considered the related Lambert's Transcendental Equation in 1758, which led to an article by [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) in 1783 that discussed the special case of\nwew\n.\nThe equation Lambert considered was\nEuler transformed this equation into the form\nBoth authors derived a series solution for their equations.\nOnce Euler had solved this equation, he considered the case a = b. Taking limits, he derived the equation\nHe then put a = 1 and obtained a convergent series solution for the resulting equation, expressing x in terms of c.\nAfter taking derivatives with respect to x and some manipulation, the standard form of the Lambert function is obtained.\nIn 1993, it was reported that the Lambert W function provides an exact solution to the quantum-mechanical [double-well Dirac delta function model](https://bliptext.com/articles/delta-potential) for equal charges —a fundamental problem in physics.\nPrompted by this, Rob Corless and developers of the [Maple](https://bliptext.com/articles/maple-software) 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.\"\nAnother example where this function is found is in [Michaelis–Menten kinetics](https://bliptext.com/articles/michaelis-menten-kinetics).\nAlthough it was widely believed that the Lambert W function cannot be expressed in terms of elementary ([Liouvillian](https://bliptext.com/articles/liouvillian-function)) functions, the first published proof did not appear until 2008.\n\n# Elementary properties, branches and range\n[[File:Lambert W Range.pdf|thumb|left|upright=1.1|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 and circle are the image of a small circle around the point\nz = 0\nz = −1/e\n.]]\n[[File:Imaginary_part_of_the_Lambert_W(n,x%2Bi_y)_for_various_branches_(n).jpg|thumb|right|upright=1.3|Plot of the imaginary part of\nWn(x + iy)\nfor branches\nn = −2, −1, 0, 1, 2\n. The plot is similar to that of the multivalued [complex logarithm](https://bliptext.com/articles/complex-logarithm) function except that the spacing between sheets is not constant and the connection of the principal sheet is different]]\nThere are countably many branches of the W function, denoted by\nWk(z)\n, for integer k;\nW0(z)\nbeing the main (or principal) branch.\nW0(z)\nis defined for all complex numbers z while\nWk(z)\nwith\nk ≠ 0\nis defined for all non-zero z.\nWith\nW0(0) = 0\nand\nlim z→0 Wk(z) = −∞\nfor all\nk ≠ 0\n.\nThe branch point for the principal branch is at\nz = −1⁄e\n, with a branch cut that extends to\n−∞\nalong the negative real axis.\nThis branch cut separates the principal branch from the two branches\nW−1\nand\nW1\n. In all branches\nWk\nwith\nk ≠ 0\n, there is a branch point at\nz = 0\nand a branch cut along the entire negative real axis.\nThe functions\nWk(z), k ∈ Z\nare all [injective](https://bliptext.com/articles/injective-function) and their ranges are disjoint.\nThe range of the entire multivalued function W is the complex plane.\nThe image of the real axis is the union of the real axis and the [quadratrix of Hippias](https://bliptext.com/articles/quadratrix-of-hippias), the parametric curve\nw = −t cot t + it\n.\n\n## Inverse\n[[File:InvertW.jpg|thumb|upright=1.3|Regions of the complex plane for which\nW(n, zez) = z\n, where\nz = x + iy\n. The darker boundaries of a particular region are included in the lighter region of the same color.\nThe point at\nis included in both the\nn = −1\n(blue) region and the\nn = 0\n(gray) region.\nHorizontal grid lines are in multiples of\nπ\n.]]\nThe range plot above also delineates the regions in the complex plane where the simple inverse relationship W(n, ze^z) = z is true.\nf = ze^z implies that there exists an n such that 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 ze^z is at the branch cut of W(n, ze^z), which means that ze^z\n≤ 0\n, except for n = 0 where it is ze^z\n≤ −1/\ne.\nDefining z = x + iy, where x and y are real, and expressing e^z in polar coordinates, it is seen that\nFor n \\neq 0, the branch cut for W(n, ze^z) is the non-positive real axis, so that\nand\nFor n = 0, the branch cut for W[n,z e^z] is the real axis with, so that the inequality becomes\nInside the regions bounded by the above, there are no discontinuous changes in W(n, ze^z), and those regions specify where the W function is simply invertible, i.e. W(n, ze^z) = z.\n\n# Calculus\n\n## Derivative\nBy [implicit differentiation](https://bliptext.com/articles/implicit-differentiation), one can show that all branches of W satisfy the [differential equation](https://bliptext.com/articles/ordinary-differential-equation)\n(W is not [differentiable](https://bliptext.com/articles/differentiable-function) for\nz = −1⁄e\n.) As a consequence, that gets the following formula for the derivative of W:\nUsing the identity\neW(z) = z⁄W(z)\n, gives the following equivalent formula:\nAt the origin we have\n\n## Integral\nThe function\nW(x)\n, and many other expressions involving\nW(x)\n, can be [integrated](https://bliptext.com/articles/integral) using the [substitution](https://bliptext.com/articles/substitution-rule)\nw = W(x)\n, i.e.\nx = wew\n(The last equation is more common in the literature but is undefined at\nx = 0\n). One consequence of this (using the fact that\nW0(e) = 1\n) is the identity\n\n# Asymptotic expansions\nThe [Taylor series](https://bliptext.com/articles/taylor-series) of\nW0\naround 0 can be found using the [Lagrange inversion theorem](https://bliptext.com/articles/lagrange-inversion-theorem) and is given by\nThe [radius of convergence](https://bliptext.com/articles/radius-of-convergence) is\n1⁄e\n, as may be seen by the [ratio test](https://bliptext.com/articles/ratio-test).\nThe function defined by this series can be extended to a [holomorphic function](https://bliptext.com/articles/holomorphic-function) defined on all complex numbers with a [branch cut](https://bliptext.com/articles/branch-cut) along the [interval](https://bliptext.com/articles/interval-mathematics) ; this [holomorphic function](https://bliptext.com/articles/holomorphic-function) defines the [principal branch](https://bliptext.com/articles/principal-branch) of the Lambert W function.\nFor large values of x,\nW0\nis asymptotic to where\nL1 = ln x\n,\nL2 = ln ln x\n, and\n[ l + m l + 1 ]\nis a non-negative [Stirling number of the first kind](https://bliptext.com/articles/stirling-numbers-of-the-first-kind).\nKeeping only the first two terms of the expansion,\nThe other real branch,\nW−1\n, defined in the interval, has an approximation of the same form as x approaches zero, with in this case\nL1 = ln(−x)\nand\nL2 = ln(−ln(−x))\n.\n\n## Integer and complex powers\nInteger powers of\nW0\nalso admit simple [Taylor](https://bliptext.com/articles/taylor-series) (or [Laurent](https://bliptext.com/articles/laurent-series)) series expansions at zero:\nMore generally, for\nr ∈ Z\n, the [Lagrange inversion formula](https://bliptext.com/articles/lagrange-inversion-theorem) gives\nwhich 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\nW0(x) / x\nwhich holds for any\nr ∈ C\nand\n.\n\n# Bounds and inequalities\nA number of non-asymptotic bounds are known for the Lambert function.\nHoorfar and Hassani showed that the following bound holds for\nx ≥ e\nThey also showed the general bound\nfor every y>1/e and x\\ge-1/e, with equality only for.\nThe bound allows many other bounds to be made, such as taking y=x+1 which gives the bound\nIn 2013 it was proven that the branch\nW−1\ncan be bounded as follows:\nRoberto Iacono and John P. Boyd enhanced the bounds as follows:\n\n# Identities\n[[File:Plot of Lambert W function of x exp x.svg|thumb|upright=1.3|A plot of\nWj(xex)\nwhere blue is for\nj = 0\nand red is for\nj = −1\n. The diagonal line represents the intervals where\nWj(xex) = x\n.]] [[File:The product logarithm Lambert W function W 2(z) plotted in the complex plane from -2-2i to 2+2i.svg|upright=1.3|alt=The product logarithm Lambert W function W 2(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The product logarithm Lambert\nW\nfunction\nW2(z)\nplotted in the complex plane from\n−2 − 2i\nto\n2 + 2i\n]] A few identities follow from the definition:\nNote that, since\nf(x) = xex\nis not [injective](https://bliptext.com/articles/injective), it does not always hold that\nW(f(x)) = x\n, much like with the [inverse trigonometric functions](https://bliptext.com/articles/inverse-trigonometric-functions).\nFor fixed\nx < 0\nand\nx ≠ −1\n, the equation\nxex = yey\nhas two real solutions in y, one of which is of course\ny = x\n. Then, for\ni = 0\nand\nx < −1\n, as well as for\ni = −1\nand\nx ∈ (−1, 0)\n,\ny = Wi(xex)\nis the other solution.\nSome other identities:\nSubstituting\n−ln x\nin the definition:\nWith Euler's iterated exponential\nh(x)\n\n# Special values\nThe following are special values of the principal branch:\nW_0(0) = 0\nW_0(e) = 1\nSpecial values of the branch\nW−1\n\n# Representations\nThe principal branch of the Lambert function can be represented by a proper integral, due to Poisson:\nAnother representation of the principal branch was found by Kalugin–Jeffrey–Corless:\nThe following [continued fraction](https://bliptext.com/articles/continued-fraction) representation also holds for the principal branch:\nAlso, if\nIn turn, if\n, then\n\n# Other formulas\n\n## Definite integrals\nThere are several useful definite integral formulas involving the principal branch of the W function, including the following:\nThe first identity can be found by writing the [Gaussian integral](https://bliptext.com/articles/gaussian-integral) in [polar coordinates](https://bliptext.com/articles/polar-coordinates).\nThe second identity can be derived by making the substitution\nu = W0(x)\n, which gives\nThus\nThe third identity may be derived from the second by making the substitution\nu = x−2\nand the first can also be derived from the third by the substitution\nz = 1⁄√2 tan x\n.\nExcept for z along the branch cut (where the integral does not converge), the principal branch of the Lambert W function can be computed by the following integral:\nwhere the two integral expressions are equivalent due to the symmetry of the integrand.\n\n## Indefinite integrals\n\n# Applications\n\n## Solving equations\nThe 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.\nThe strategy is to convert such an equation into one of the form\nzez = w\nand then to solve for z using the W function.\nFor example, the equation\n(where x is an unknown real number) can be solved by rewriting it as\nThis last equation has the desired form and the solutions for real x are:\nand thus:\nGenerally, the solution to is:\nwhere a, b, and c are complex constants, with b and c not equal to zero, and the W function is of any integer order.\n\n## Viscous flows\nGranular 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:\nwhere\nH(x)\nis 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.\nIn [pipe flow](https://bliptext.com/articles/pipe-flow), the Lambert W function is part of the explicit formulation of the [Colebrook equation](https://bliptext.com/articles/colebrook-equation) for finding the [Darcy friction factor](https://bliptext.com/articles/darcy-friction-factor).\nThis factor is used to determine the pressure drop through a straight run of pipe when the flow is [turbulent](https://bliptext.com/articles/turbulent).\n\n## Time-dependent flow in simple branch hydraulic systems\nThe principal branch of the Lambert W function is employed in the field of [mechanical engineering](https://bliptext.com/articles/mechanical-engineering), in the study of time dependent transfer of [Newtonian fluids](https://bliptext.com/articles/newtonian-fluid) between two reservoirs with varying free surface levels, using centrifugal pumps.\nThe Lambert W function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes:\nwhere Q_i is the initial flow rate and t is time.\n\n## Neuroimaging\nThe Lambert W function is employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain [voxel](https://bliptext.com/articles/voxel), to the corresponding blood oxygenation level dependent (BOLD) signal.\n\n## Chemical engineering\nThe Lambert W function is employed in the field of chemical engineering for modeling the porous electrode film thickness in a [glassy carbon](https://bliptext.com/articles/glassy-carbon) based [supercapacitor](https://bliptext.com/articles/supercapacitor) for electrochemical energy storage.\nThe Lambert W function provides an exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.\n\n## Crystal growth\nIn the crystal growth, the negative principal of the Lambert W-function can be used to calculate the distribution coefficient, k, and solute concentration in the melt, C_L, from the [Scheil equation](https://bliptext.com/articles/scheil-equation):\n\n## Materials science\nThe Lambert W function is employed in the field of [epitaxial film growth](https://bliptext.com/articles/epitaxy) for the determination of the critical [dislocation](https://bliptext.com/articles/dislocation) onset film thickness.\nThis 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.\nPrior to application of Lambert W for this problem, the critical thickness had to be determined via solving an implicit equation.\nLambert W turns it in an explicit equation for analytical handling with ease.\n\n## Porous media\nThe 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.\nThe 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.\n\n## Bernoulli numbers and Todd genus\nThe equation (linked with the generating functions of [Bernoulli numbers](https://bliptext.com/articles/bernoulli-number) and [Todd genus](https://bliptext.com/articles/genus-of-a-multiplicative-sequence)):\ncan be solved by means of the two real branches\nW0\nand\nW−1\nThis application shows that the branch difference of the W function can be employed in order to solve other transcendental equations.\n\n## Statistics\nThe centroid of a set of histograms defined with respect to the symmetrized Kullback–Leibler divergence (also called the Jeffreys divergence ) has a closed form using the Lambert W function.\n\n## Pooling of tests for infectious diseases\nSolving for the optimal group size to pool tests so that at least one individual is infected involves the Lambert\nW\nfunction.\n\n## Exact solutions of the Schrödinger equation\nThe 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](https://bliptext.com/articles/inverse-square-root-potential) – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions.\nThe 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\nThe 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](https://bliptext.com/articles/delta-potential).\n\n## Exact solution of QCD coupling constant\nIn [Quantum chromodynamics](https://bliptext.com/articles/quantum-chromodynamics), the [quantum field theory](https://bliptext.com/articles/quantum-field-theory) of the [Strong interaction](https://bliptext.com/articles/strong-interaction), the [coupling constant](https://bliptext.com/articles/coupling-constant) is computed perturbatively, the order n corresponding to [Feynman diagrams](https://bliptext.com/articles/feynman-diagrams) including n quantum loops.\nThe first order,\nn = 1\n, solution is exact (at that order) and analytical.\nAt higher orders,\nn > 1\n, there is no exact and analytical solution and one typically uses an [iterative method](https://bliptext.com/articles/iterative-method) to furnish an approximate solution.\nHowever, for second order,\nn = 2\n, the Lambert function provides an exact (if non-analytical) solution.\n\n## Exact solutions of the Einstein vacuum equations\nIn the [Schwarzschild metric](https://bliptext.com/articles/schwarzschild-metric) solution of the Einstein vacuum equations, the W function is needed to go from the [Eddington–Finkelstein coordinates](https://bliptext.com/articles/eddington-finkelstein-coordinates) to the Schwarzschild coordinates.\nFor this reason, it also appears in the construction of the [Kruskal–Szekeres coordinates](https://bliptext.com/articles/kruskal-szekeres-coordinates).\n\n## Resonances of the delta-shell potential\nThe s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert W function.\n\n## Thermodynamic equilibrium\nIf a reaction involves reactants and products having [heat capacities](https://bliptext.com/articles/heat-capacity) that are constant with temperature then the equilibrium constant K obeys\nfor some constants a, b, and c. When c (equal to\nΔCp⁄R\n) is not zero the value or values of T can be found where K equals a given value as follows, where L can be used for\nln T\n.\nIf a and c have the same sign there will be either two solutions or none (or one if the argument of W is exactly\n−1⁄e\n). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.\n\n## Phase separation of polymer mixtures\nIn the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to the [Edmond-Ogston model](https://bliptext.com/articles/edmond-ogston-model), the solutions for binodal and tie-lines are formulated in terms of Lambert W functions.\n\n## Wien's displacement law in a D-dimensional universe\nWien's displacement law is expressed as.\nWith and, where \\rho_{T} is the spectral energy energy density, one finds , where D is the number of degrees of freedom for spatial translation.\nThe solution shows that the spectral energy density is dependent on the dimensionality of the universe.\n\n## AdS/CFT correspondence\nThe classical finite-size corrections to the dispersion relations of [giant magnons](https://bliptext.com/articles/giant-magnon), single spikes and [GKP strings](https://bliptext.com/articles/gkp-string) can be expressed in terms of the Lambert W function.\n\n## Epidemiology\nIn the\nt → ∞\nlimit of the [SIR model](https://bliptext.com/articles/compartmental-models-in-epidemiology), the proportion of susceptible and recovered individuals has a solution in terms of the Lambert W function.\n\n## Determination of the time of flight of a projectile\nThe total time of the journey of a projectile which experiences air resistance proportional to its velocity [can be determined](https://bliptext.com/articles/projectile-motion) in exact form by using the Lambert\nW\nfunction.\n\n## Electromagnetic surface wave propagation\nThe 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\nu ln u = v\n(where u and v clump together the geometrical and physical factors of the problem), which is solved by the Lambert W function.\nThe first solution to this problem, due to Sommerfeld circa 1898, already contained an iterative method to determine the value of the Lambert W function.\n\n## Orthogonal trajectories of real ellipses\nThe family of ellipses centered at (0, 0) is parameterized by eccentricity \\varepsilon.\nThe orthogonal trajectories of this family are given by the differential equation whose general solution is the family y^2=.\n\n# Generalizations\nThe standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:\nwhere\na0\n, c and r are real constants.\nThe solution is\nGeneralizations of the Lambert W function include:\n

An application to [general relativity](https://bliptext.com/articles/general-relativity) and [quantum mechanics](https://bliptext.com/articles/quantum-mechanics) ([quantum gravity](https://bliptext.com/articles/quantum-gravity)) in lower dimensions, in fact a link (unknown prior to 2007 ) between these two areas, where the right-hand side of is replaced by a quadratic polynomial in\nx\nwhere\nr1\nand\nr2\nare real distinct constants, the roots of the quadratic polynomial.\nHere, the solution is a function which has a single argument x but the terms like\nri\nand\na0\nare parameters of that function.\nIn this respect, the generalization resembles the [hypergeometric](https://bliptext.com/articles/hypergeometric) function and the [Meijer G function](https://bliptext.com/articles/meijer-g-function) but it belongs to a different class of functions.\nWhen\nr1 = r2\n, both sides of can be factored and reduced to and thus the solution reduces to that of the standard W function.\nEquation expresses the equation governing the [dilaton](https://bliptext.com/articles/dilaton) field, from which is derived the metric of the [[R = T model|\nR = T\n]] 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](https://bliptext.com/articles/delta-potential) for unequal charges in one dimension.\n
Analytical solutions of the eigenenergies of a special case of the quantum mechanical [three-body problem](https://bliptext.com/articles/euler-s-three-body-problem), namely the (three-dimensional) [hydrogen molecule-ion](https://bliptext.com/articles/hydrogen-molecule-ion).\nHere the right-hand side of is replaced by a ratio of infinite order polynomials in x:\nwhere\nri\nand\nsi\nare distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Equation with its specialized cases expressed in and is related to a large class of [delay differential equations](https://bliptext.com/articles/delay-differential-equation).\n[G. H. Hardy](https://bliptext.com/articles/g-h-hardy)'s notion of a \"false derivative\" provides exact multiple roots to special cases of.\n

\nApplications of the Lambert W function in fundamental physical problems are not exhausted even for the standard case expressed in as seen recently in the area of [atomic, molecular, and optical physics](https://bliptext.com/articles/atomic-molecular-and-optical-physics).\n\n# Plots\n\n# Numerical evaluation\nThe W function may be approximated using [Newton's method](https://bliptext.com/articles/newton-s-method), with successive approximations to\nw = W(z)\n(so\nz = wew\n) being\nThe W function may also be approximated using [Halley's method](https://bliptext.com/articles/halley-s-method),\ngiven in Corless et al. to compute W.\nFor real x \\ge -1/e, it may be approximated by the quadratic-rate recursive formula of R. Iacono and J.P. Boyd:\nLajos Lóczi proves that by using this iteration with an appropriate starting value w_0 (x),\none can determine the maximum number of iteration steps in advance for any precision:\nToshio Fukushima has presented a fast method for approximating the real valued parts of the principal and secondary branches of the W function without using any iteration.\nIn this method the W function is evaluated as a conditional switch of [rational functions](https://bliptext.com/articles/rational-functions) on transformed varibles:\nwhere x, u, y and v are transformations of z:\nHere X_k(x), U_k(u), Y_k(y), and V_k(v) are rational functions whose coefficients for different k-values are listed in the referenced paper together with the z_k values that determine the subdomains.\nWith higher degree polynomials in these rational functions the method can approximate the W function more accurately.\nFor example, when, W_0(z) can be approximated to 24 bits of accuracy on 64-bit floating point values as where x is defined with the transformation above and the coefficients P_i and Q_i are given in the table below.\nFukushima also offers an approximation with 50 bits of accuracy on 64-bit floats that uses 8th- and 7th-degree polynomials.\n\n# Software\nThe Lambert W function is implemented in many programming languages.\nSome of them are listed below:\nC++ code for all the branches of the complex Lambert W function is also available on the homepage of István Mező.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:53:08.890Z","updated_at":"2025-01-25T17:53:08.890Z","search_vector":"'function':3 'lambert':1 'w':2","search_content":" \n \n\n# \n[[File:The product logarithm Lambert W function plotted in the complex plane from -2-2i to 2+2i.svg|upright=1.3|alt=The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i|thumb|The product logarithm Lambert W function plotted in the complex plane from\n−2 − 2i\nto\n2 + 2i\n]] [[File:Mplwp lambert W branches.svg|thumb|upright=1.3|The graph of\ny = W(x)\nfor real\nx −4\n. The upper branch (blue) with\ny ≥ −1\nis the graph of the function\nW0\n(principal branch), the lower branch (magenta) with\ny ≤ −1\nis the graph of the function\nW−1\n. The minimum value of\nx\nis at\n]]\nIn [mathematics](https://bliptext.com/articles/mathematics), the **Lambert W function**, also called the **omega function** or **product logarithm**, is a [multivalued function](https://bliptext.com/articles/multivalued-function), namely the [branches](https://bliptext.com/articles/branch-point) of the [converse relation](https://bliptext.com/articles/converse-relation) of the function\nf(w) = wew\n, where w is any [complex number](https://bliptext.com/articles/complex-number) and\new\nis the [exponential function](https://bliptext.com/articles/exponential-function).\nThe function is named after [Johann Lambert](https://bliptext.com/articles/johann-heinrich-lambert), who considered a related problem in 1758.\nBuilding on Lambert's work, [Leonhard Euler](https://bliptext.com/articles/leonhard-euler) described the W function per se in 1783.\nFor each integer\nk\nthere is one branch, denoted by\nWk(z)\n, which is a complex-valued function of one complex argument.\nW0\nis known as the [principal branch](https://bliptext.com/articles/principal-branch).\nThese functions have the following property: if\nz\nand\nw\nare any complex numbers, then\nholds if and only if\nWhen dealing with real numbers only, the two branches\nW0\nand\nW−1\nsuffice: for real numbers\nx\nand\ny\nthe equation\ncan be solved for\ny\nonly if\nx ≥ −1⁄e\ny = W0(x)\nif\nx ≥ 0\nand the two values\ny = W0(x)\nand\ny = W−1(x)\nif\n−1⁄e ≤ x 1/e and x\\ge-1/e, with equality only for.\nThe bound allows many other bounds to be made, such as taking y=x+1 which gives the bound\nIn 2013 it was proven that the branch\nW−1\ncan be bounded as follows:\nRoberto Iacono and John P. Boyd enhanced the bounds as follows:\n\n# Identities\n[[File:Plot of Lambert W function of x exp x.svg|thumb|upright=1.3|A plot of\nWj(xex)\nwhere blue is for\nj = 0\nand red is for\nj = −1\n. The diagonal line represents the intervals where\nWj(xex) = x\n.]] [[File:The product logarithm Lambert W function W 2(z) plotted in the complex plane from -2-2i to 2+2i.svg|upright=1.3|alt=The product logarithm Lambert W function W 2(z) plotted in the complex plane from -2-2i to 2+2i|thumb|The product logarithm Lambert\nW\nfunction\nW2(z)\nplotted in the complex plane from\n−2 − 2i\nto\n2 + 2i\n]] A few identities follow from the definition:\nNote that, since\nf(x) = xex\nis not [injective](https://bliptext.com/articles/injective), it does not always hold that\nW(f(x)) = x\n, much like with the [inverse trigonometric functions](https://bliptext.com/articles/inverse-trigonometric-functions).\nFor fixed\nx 1\n, there is no exact and analytical solution and one typically uses an [iterative method](https://bliptext.com/articles/iterative-method) to furnish an approximate solution.\nHowever, for second order,\nn = 2\n, the Lambert function provides an exact (if non-analytical) solution.\n\n## Exact solutions of the Einstein vacuum equations\nIn the [Schwarzschild metric](https://bliptext.com/articles/schwarzschild-metric) solution of the Einstein vacuum equations, the W function is needed to go from the [Eddington–Finkelstein coordinates](https://bliptext.com/articles/eddington-finkelstein-coordinates) to the Schwarzschild coordinates.\nFor this reason, it also appears in the construction of the [Kruskal–Szekeres coordinates](https://bliptext.com/articles/kruskal-szekeres-coordinates).\n\n## Resonances of the delta-shell potential\nThe s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert W function.\n\n## Thermodynamic equilibrium\nIf a reaction involves reactants and products having [heat capacities](https://bliptext.com/articles/heat-capacity) that are constant with temperature then the equilibrium constant K obeys\nfor some constants a, b, and c. When c (equal to\nΔCp⁄R\n) is not zero the value or values of T can be found where K equals a given value as follows, where L can be used for\nln T\n.\nIf a and c have the same sign there will be either two solutions or none (or one if the argument of W is exactly\n−1⁄e\n). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.\n\n## Phase separation of polymer mixtures\nIn the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to the [Edmond-Ogston model](https://bliptext.com/articles/edmond-ogston-model), the solutions for binodal and tie-lines are formulated in terms of Lambert W functions.\n\n## Wien's displacement law in a D-dimensional universe\nWien's displacement law is expressed as.\nWith and, where \\rho_{T} is the spectral energy energy density, one finds , where D is the number of degrees of freedom for spatial translation.\nThe solution shows that the spectral energy density is dependent on the dimensionality of the universe.\n\n## AdS/CFT correspondence\nThe classical finite-size corrections to the dispersion relations of [giant magnons](https://bliptext.com/articles/giant-magnon), single spikes and [GKP strings](https://bliptext.com/articles/gkp-string) can be expressed in terms of the Lambert W function.\n\n## Epidemiology\nIn the\nt → ∞\nlimit of the [SIR model](https://bliptext.com/articles/compartmental-models-in-epidemiology), the proportion of susceptible and recovered individuals has a solution in terms of the Lambert W function.\n\n## Determination of the time of flight of a projectile\nThe total time of the journey of a projectile which experiences air resistance proportional to its velocity [can be determined](https://bliptext.com/articles/projectile-motion) in exact form by using the Lambert\nW\nfunction.\n\n## Electromagnetic surface wave propagation\nThe 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\nu ln u = v\n(where u and v clump together the geometrical and physical factors of the problem), which is solved by the Lambert W function.\nThe first solution to this problem, due to Sommerfeld circa 1898, already contained an iterative method to determine the value of the Lambert W function.\n\n## Orthogonal trajectories of real ellipses\nThe family of ellipses centered at (0, 0) is parameterized by eccentricity \\varepsilon.\nThe orthogonal trajectories of this family are given by the differential equation whose general solution is the family y^2=.\n\n# Generalizations\nThe standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form:\nwhere\na0\n, c and r are real constants.\nThe solution is\nGeneralizations of the Lambert W function include:\n \n An application to [general relativity](https://bliptext.com/articles/general-relativity) and [quantum mechanics](https://bliptext.com/articles/quantum-mechanics) ([quantum gravity](https://bliptext.com/articles/quantum-gravity)) in lower dimensions, in fact a link (unknown prior to 2007 ) between these two areas, where the right-hand side of is replaced by a quadratic polynomial in\nx\nwhere\nr1\nand\nr2\nare real distinct constants, the roots of the quadratic polynomial.\nHere, the solution is a function which has a single argument x but the terms like\nri\nand\na0\nare parameters of that function.\nIn this respect, the generalization resembles the [hypergeometric](https://bliptext.com/articles/hypergeometric) function and the [Meijer G function](https://bliptext.com/articles/meijer-g-function) but it belongs to a different class of functions.\nWhen\nr1 = r2\n, both sides of can be factored and reduced to and thus the solution reduces to that of the standard W function.\nEquation expresses the equation governing the [dilaton](https://bliptext.com/articles/dilaton) field, from which is derived the metric of the [[R = T model|\nR = T\n]] 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](https://bliptext.com/articles/delta-potential) for unequal charges in one dimension.\n \n Analytical solutions of the eigenenergies of a special case of the quantum mechanical [three-body problem](https://bliptext.com/articles/euler-s-three-body-problem), namely the (three-dimensional) [hydrogen molecule-ion](https://bliptext.com/articles/hydrogen-molecule-ion).\nHere the right-hand side of is replaced by a ratio of infinite order polynomials in x:\nwhere\nri\nand\nsi\nare distinct real constants and x is a function of the eigenenergy and the internuclear distance R. Equation with its specialized cases expressed in and is related to a large class of [delay differential equations](https://bliptext.com/articles/delay-differential-equation).\n[G. H. Hardy](https://bliptext.com/articles/g-h-hardy)'s notion of a \"false derivative\" provides exact multiple roots to special cases of.\n \n \nApplications of the Lambert W function in fundamental physical problems are not exhausted even for the standard case expressed in as seen recently in the area of [atomic, molecular, and optical physics](https://bliptext.com/articles/atomic-molecular-and-optical-physics).\n\n# Plots\n\n# Numerical evaluation\nThe W function may be approximated using [Newton's method](https://bliptext.com/articles/newton-s-method), with successive approximations to\nw = W(z)\n(so\nz = wew\n) being\nThe W function may also be approximated using [Halley's method](https://bliptext.com/articles/halley-s-method),\ngiven in Corless et al. to compute W.\nFor real x \\ge -1/e, it may be approximated by the quadratic-rate recursive formula of R. Iacono and J.P. Boyd:\nLajos Lóczi proves that by using this iteration with an appropriate starting value w_0 (x),\none can determine the maximum number of iteration steps in advance for any precision:\nToshio Fukushima has presented a fast method for approximating the real valued parts of the principal and secondary branches of the W function without using any iteration.\nIn this method the W function is evaluated as a conditional switch of [rational functions](https://bliptext.com/articles/rational-functions) on transformed varibles:\nwhere x, u, y and v are transformations of z:\nHere X_k(x), U_k(u), Y_k(y), and V_k(v) are rational functions whose coefficients for different k-values are listed in the referenced paper together with the z_k values that determine the subdomains.\nWith higher degree polynomials in these rational functions the method can approximate the W function more accurately.\nFor example, when, W_0(z) can be approximated to 24 bits of accuracy on 64-bit floating point values as where x is defined with the transformation above and the coefficients P_i and Q_i are given in the table below.\nFukushima also offers an approximation with 50 bits of accuracy on 64-bit floats that uses 8th- and 7th-degree polynomials.\n\n# Software\nThe Lambert W function is implemented in many programming languages.\nSome of them are listed below:\nC++ code for all the branches of the complex Lambert W function is also available on the homepage of István Mező.","revisionCount":0},"hasMore":false,"nextCursor":null}