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 range plot above also delineates the regions in the complex plane where the simple inverse relationship ⁠

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.

This factor is used to determine the pressure drop through a straight run of pipe when the flow is turbulent.

[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.

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.

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.

The product logarithm Lambert W function plotted in the complex plane from −2 − 2i to 2 + 2i
The product logarithm Lambert W function plotted in the complex plane from −2 − 2 i to 2 + 2 i
The graph of y = W ( x ) for real x < 6 and y > −4 . The upper branch (blue) with y ≥ −1 is the graph of the function W 0 (principal branch), the lower branch (magenta) with y ≤ −1 is the graph of the function W −1 . The minimum value of x is at {−1/ e , −1}
Main branch of the Lambert W function in the complex plane, plotted with domain coloring . Note the branch cut along the negative real axis, ending at − ⁠ 1 / e ⁠ .
The modulus of the principal branch of the Lambert W function, colored according to arg W ( z )
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 z = 0 ; the red curves are the image of a small circle around the point z = −1/e .
Plot of the imaginary part of W n ( x + iy ) for branches n = −2, −1, 0, 1, 2 . The plot is similar to that of the multivalued complex logarithm function except that the spacing between sheets is not constant and the connection of the principal sheet is different
Regions of the complex plane for which W ( n , ze z ) = z , where z = x + iy . The darker boundaries of a particular region are included in the lighter region of the same color. The point at {−1, 0} is included in both the n = −1 (blue) region and the n = 0 (gray) region. Horizontal grid lines are in multiples of π .
A plot of W j ( xe x ) where blue is for j = 0 and red is for j = −1 . The diagonal line represents the intervals where W j ( xe x ) = x .
The product logarithm Lambert W function W 2(z) plotted in the complex plane from -2-2i to 2+2i
The product logarithm Lambert W function W 2 ( z ) plotted in the complex plane from −2 − 2 i to 2 + 2 i