In mathematics, Lambert's W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the inverse function of Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... Johann Heinrich Lambert Johann Heinrich Lambert (August 26, 1728 – September 25, 1777), was a mathematician, physicist and astronomer. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

where e^w is the exponential function and w is any complex number. Lambert's function is usually denoted W(z). For every complex number z, we have The exponential function is one of the most important functions in mathematics. ...

Since the function f is not injective in (−∞, 0), the function W is multivalued in [−1/e, 0). If we restrict to real arguments x ≥ −1/e and demand w ≥ −1, then a single-valued function W₀(x) is defined, whose graph is shown. We have W₀(0) = 0 and W₀(−1/e) = −1. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... This diagram does not represent a true function, because the element 3 in X is associated with two elements, b and c, in Y. In mathematics, a multivalued function is a total relation; i. ...

The Lambert W function cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials and also occurs in the solution of time-delayed differential equations, such as y'(t) = a y(t − 1). In mathematics, several functions are important enough to deserve their own name. ... Combinatorics is a odd branch of mathematics that studies collections (usually finite) then constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding largest, smallest, or optimal objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics). ... A tree with 6 vertices and 5 edges In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. ...

By implicit differentiation, one can show that W satisfies the differential equation In mathematics, to give an implicit function f is to give the graph of a function, as a relation. ... An ordinary differential equation, or ODE, is an equation depending on a single spatial variable. ...

The Taylor series of W₀ around 0 can be found using the Lagrange inversion theorem and is given by As the degree of the Taylor series rises, it approaches the correct function. ... In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function. ...

The radius of convergence is 1/e, as may be seen by the ratio test. The function defined by this series can be extended to a holomorphic function defined on all complex numbers with a branch cut along the interval (−∞, −1/e]; this holomorphic function is defines the principal branch of the Lambert W function. In mathematics, the radius of convergence of a power series where the center a and the coefficients cn are complex numbers (which may, in particular, be real numbers) is the nonnegative quantity r (which may be a real number or ∞) such that the series converges if and diverges if In... In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In complex analysis, a branch point may be thought of informally as a point z0 at which a multiple_valued function changes values when one winds once around z0. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, a principal branch is a function which selects one branch, or slice, of a multi-valued function. ...

Many equations involving exponentials can be solved using the W function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like x e^x, at which point the W function provides the solution. For instance, to solve the equation 2^t = 5t, we divide by 2^t to get 1 = 5t e^−ln(2)t, then divide by 5 and multiply by −ln(2) to get −ln(2)/5 = −ln(2)t e^−ln(2)t. Now application of the W function yields −ln(2)t = W(−ln(2)/5), i.e. t = −W(−ln(2)/5) / ln(2). In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...

Similar techniques show that

x^x = z

has solution

or, equivalently,

Whenever the complex infinite exponential,

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

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

The function W(x), and many expressions involving W(x), can be integrated using the substitution w = W(x), i.e. x = w e^w: Integration may be any of the following: In the most general sense, integration may be any bringing together of things: the integration of two or more economies, cultures, religions (usually called syncretism), etc. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

Special values

(the Omega constant)

The Omega constant is the value of W(1) where W is Lamberts W function. ...

Evaluation algorithm

The W function may be evaluated using the recurrence relation Recurrent redirects here; for the meaning of recurrent in contemporary hit radio, see Recurrent rotation. ...

given in Corless et. al. to compute W. Together with the evaluation error estimate given in Chapeau-Belandeau and Monir, the following Python code implements this:

 import math class Error(Exception): pass def lambertW(x, prec=1e-12): w = 0 for i in xrange(100): wTimesExpW = w*math.exp(w) wPlusOneTimesExpW = (w+1)*math.exp(w) if prec>abs((x-wTimesExpW)/wPlusOneTimesExpW): break w = w-(wTimesExpW-x)/( wPlusOneTimesExpW-(w+2)*(wTimesExpW-x)/(2*w+2)) else: raise Error, "W doesn't converge fast enough for %f"%x return w 

This computes the principal branch for x > 1 / e. It could be improved by giving better initial estimates.

The following closed form approximation may be used by itself when less accuracy is needed, or to give an excellent initial estimate to the above code, which then may need only a few iterations:

 double desy_lambert_W(double x) { double lx1; if (x <= 500.0) { lx1 = log(x + 1.0); return 0.665 * (1 + 0.0195 * lx1) * lx1 + 0.04; } return log(x - 4.0) - (1.0 - 1.0/log(x)) * log(log(x)); } 

(from http://www.desy.de/~t00fri/qcdins/texhtml/lambertw/)

References and external links

• Corless et al. "On the Lambert W function" Adv. Computational Maths. 5, 329 - 359 (1996) (PostScript)
• Chapeau-Blondeau, F. and Monir, A: "Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2", IEEE Trans. Signal Processing, 50(9), 2002
• MathWorld - Lambert W-Function
• Lambert function from Wolfram's function site.
• Francis et al. "Quantitative General Theory for Periodic Breathing" Circulation 102 (18): 2214. (2000). Use of Lambert function to solve delay-differential dynamics in human disease.
• Extreme Mathematics. Monographs on the Lambert W function, its numerical approximation and generalizations for W-like inverses of transcendental forms with repeated exponential towers.