Lambert W function

[citation needed] For each integer k there is one branch, denoted by Wk(z), which is a complex-valued function of one complex argument.

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.

(Technical note: like the complex logarithm, it is multivalued and thus W is described as a converse relation rather than inverse function.)

The function W(x), and many other expressions involving W(x), can be integrated using the substitution w = W(x), i.e. x = wew: (The last equation is more common in the literature but is undefined at x = 0).

One consequence of this (using the fact that W0(e) = 1) is the identity The Taylor series of W0 around 0 can be found using the Lagrange inversion theorem and is given by The radius of convergence is ⁠1/e⁠, as may be seen by the ratio test.

[3] Integer powers of W0 also admit simple Taylor (or Laurent) series expansions at zero: More generally, for r ∈ Z, the Lagrange inversion formula gives which is, in general, a Laurent series of order r. Equivalently, the latter can be written in the form of a Taylor expansion of powers of W0(x) / x: which holds for any r ∈ C and |x| < ⁠1/e⁠.

Some other identities:[13] Substituting −ln x in the definition:[15] With Euler's iterated exponential h(x): The following are special values of the principal branch:

The Lambert W function is used to solve equations in which the unknown quantity occurs both in the base and in the exponent, or both inside and outside of a logarithm.

In pipe flow, the Lambert W function is part of the explicit formulation of the Colebrook equation for finding the Darcy friction factor.

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

[23] The Lambert W function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes:

[24] The Lambert W function is employed in the field of chemical engineering for modeling the porous electrode film thickness in a glassy carbon based supercapacitor for electrochemical energy storage.

[25][26] In the crystal growth, the negative principal of the Lambert W-function can be used to calculate the distribution coefficient,

Prior to application of Lambert W for this problem, the critical thickness had to be determined via solving an implicit equation.

[31] The equation (linked with the generating functions of Bernoulli numbers and Todd genus): can be solved by means of the two real branches W0 and W−1: This application shows that the branch difference of the W function can be employed in order to solve other transcendental equations.

[34] Solving for the optimal group size to pool tests so that at least one individual is infected involves the Lambert W function.

[35][36][37] The Lambert W function appears in a quantum-mechanical potential, which affords the fifth – next to those of the harmonic oscillator plus centrifugal, the Coulomb plus inverse square, the Morse, and the inverse square root potential – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions.

is computed perturbatively, the order n corresponding to Feynman diagrams including n quantum loops.

The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert W function.

In the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to the Edmond-Ogston model, the solutions for binodal and tie-lines are formulated in terms of Lambert W functions.

[42] The classical finite-size corrections to the dispersion relations of giant magnons, single spikes and GKP strings can be expressed in terms of the Lambert W function.

[43][44] In the t → ∞ limit of the SIR model, the proportion of susceptible and recovered individuals has a solution in terms of the Lambert W function.

[45] The total time of the journey of a projectile which experiences air resistance proportional to its velocity can be determined in exact form by using the Lambert W function.

[46] The transcendental equation that appears in the determination of the propagation wave number of an electromagnetic axially symmetric surface wave (a low-attenuation single TM01 mode) propagating in a cylindrical metallic wire gives rise to an equation like u ln u = v (where u and v clump together the geometrical and physical factors of the problem), which is solved by the Lambert W function.

The first solution to this problem, due to Sommerfeld circa 1898, already contained an iterative method to determine the value of the Lambert W function.

The standard Lambert W function expresses exact solutions to transcendental algebraic equations (in x) of the form: where a0, c and r are real constants.

Generalizations of the Lambert W function[48][49][50] include: where r1 and r2 are real distinct constants, the roots of the quadratic polynomial.

G. H. Hardy's notion of a "false derivative" provides exact multiple roots to special cases of (3).

, it may be approximated by the quadratic-rate recursive formula of R. Iacono and J.P. Boyd:[12] Lajos Lóczi proves[55] that by using this iteration with an appropriate starting value

Toshio 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.