Lambert W function - Citizendia

The graph of W₀(x) for −1/e ≤ x ≤ 4

In mathematics, The Lambert W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the inverse function of f(w) = we^w where e^w is the natural exponential function and w is any complex number. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) We will denote the function here by W. For every complex number z, we have

z = W (z) e W (z) .

Since the function f is not injective, the function W is multivalued (except at 0). In Mathematics, a multivalued function (shortly multifunction, other names set-valued function, set-valued map, multi-valued map If we restrict to real arguments x and demand w real, then the function is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0); the additional constraint w ≥ −1 defines a single-valued function W₀(x), whose graph is shown. We have W₀(0) = 0 and W₀(−1/e) = −1. The alternate branch on [−1/e, 0) with w ≤ −1 is denoted W₋₁(x) and decreases from W₋₁(−1/e) = −1 to W₋₁(0⁻) = −∞.

The Lambert W function cannot be expressed in terms of elementary functions. This article discusses the concept of elementary functions in differential algebra It is useful in combinatorics, for instance in the enumeration of trees. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Graph theory, a tree is a graph in which any two vertices are connected by exactly one path. 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).

Lambert W function in the complex plane

1 Differentiation and integration
2 Taylor series
3 Applications
- 3.1 Examples
4 Special values
5 Plots
6 Evaluation algorithm
7 History
8 References and external links

Differentiation and integration

By implicit differentiation, one can show that W satisfies the differential equation

$z(1+W)\frac{dW}{dz}=W\quad\mathrm{for\ }z\neq -1/e \,,$

and hence:

$\frac{dW}{dz}=\frac{W(z)}{z(1 + W(z))}\quad\mathrm{for\ }z\neq -1/e \,.$

The function W(x), and many expressions involving W(x), can be integrated using the substitution w = W(x), i. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires e. x = w e^w:

$\int W(x)\, dx = x \left( W(x) - 1 + \frac{1}{W(x)} \right) + C$

Taylor series

The Taylor series of W₀ around 0 can be found using the Lagrange inversion theorem and is given by

$W_0 (x) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}\ x^n = x - x^2 + \frac{3}{2}x^3 - \frac{8}{3}x^4 + \frac{125}{24}x^5 - \cdots$

The radius of convergence is 1/e, as may be seen by the ratio test. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the Inverse In Mathematics, the radius of convergence of a Power series is a non-negative quantity either a real number or \scriptstyle \infty that represents a In Mathematics, the ratio test is a test (or "criterion" for the convergence of a series \sum_{n=0}^\infty a_n 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 defines the principal branch of the Lambert W function. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In the mathematical field of Complex analysis, a branch point may be informally thought of as a point z 0 at which a " multi-valued In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, a principal branch is a function which selects one branch or "slice" of a Multi-valued function.

Applications

Many equations involving exponentials can be solved using the W function. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent The general strategy is to move all instances of the unknown to one side of the equation and make it look like Y = Xe^X at which point the W function provides the solution.

In other words :

$X = Y e ^ Y \; \Longleftrightarrow \; Y = W(X)$

Examples

Example 1

$2^t = 5 t\,$

$\Rightarrow 1 = \frac{5 t}{2^t}\,$

$\Rightarrow 1 = 5 t \, e^{-t \ln 2}\,$

$\Rightarrow \frac{1}{5} = t \, e^{-t \ln 2}\,$

$\Rightarrow \frac{- \, \ln 2}{5} = ( - \, t \, \ln 2 ) \, e^{( -t \ln 2 )}\,$

$\Rightarrow -t \ln 2 = W \left ( \frac{- \ln 2}{5} \right )\,$

$\Rightarrow t = \frac{- W \left ( \frac{- \ln 2}{5} \right )}{\ln 2}\,$

More generally, the equation

$~p^{a x + b} = c x + d$

where

$p > 0 \and c,d \neq 0$

can be transformed via the substitution

$-t = a x + \frac{a d}{c}$

into

$t p^t = R = -\frac{a}{c} p^{b-\frac{a d}{c}}$

giving

$t = \frac{W(R\ln p)}{\ln p}$

which yields the final solution

$x = -\frac{W(-\frac{a\ln p}{c}\,p^{b-\frac{a d}{c}})}{a\ln p} - \frac{d}{c}$

Example 2

Similar techniques show that

$x^x=z\, ,$

has solution

$x=\frac{\ln(z)}{W(\ln z)}\, ,$

or, equivalently,

$x=\exp\left(W(\ln(z))\right).$

Example 3

Whenever the complex infinite exponential tetration

$z^{z^{z^{\cdot^{\cdot^{\cdot}}}}} \!$

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

$c=\frac{W(-\ln(z))}{-\ln(z)}$

where ln(z) denotes the principal branch of the complex log function. In Mathematics, tetration (also known as hyper -4

Example 4

Solutions for

$x \log_b \left(x\right) = a$

have the form

$x = \frac{a \ln(b)}{W(a \ln(b))}$

Example 5

The solution for the current in a series diode/resistor circuit can also be written in terms of the Lambert W. Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. Dioden2jpg|thumb|right|150px|Figure 2 Various semiconductor diodes |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| | See diode modeling. In Electronics, diode modeling refers to the mathematical models used to approximate the actual behavior of real diodes to enable calculations and circuit analysis

Example 6

The delay differential equation

$\dot{y}(t) = ay(t-1)$

has characteristic equation $λ = a e - λ$ , leading to $λ = W k (a)$ and $y(t)=e^{W_k(a)t}$ , where $k$ is the branch index. In Mathematics, delay differential equations (DDEs are a type of Differential equation in which the derivative of the unknown function at a certain time is given In Mathematics, delay differential equations (DDEs are a type of Differential equation in which the derivative of the unknown function at a certain time is given If $a$ is real, only $W 0 (a)$ need be considered.

Special values

$W\left(-\frac{\pi}{2}\right) = \frac{i\pi}{2}$

$W\left(-\frac{\ln 2}{2}\right)= -\ln 2$

$W\left(-{1\over e}\right) = -1$

$W(0) = 0\,$

$W(1) = \Omega\,$ (the Omega constant)

$W(e) = 1\,$

Plots

Plots of the LambertW function on the complex plane
z = Re(W₀(x + i y))	z = Im(W₀(x + i y))	z = \|W₀(x + i y)\|

Evaluation algorithm

The W function may be evaluated using the recurrence relation

$w_{j+1}=w_j-\frac{w_j e^{w_j}-z}{e^{w_j}(w_j+1)-\frac{(w_j+2)(w_je^{w_j}-z)} {2w_j+2}}$

given in Corless et al. The Omega constant is a Mathematical constant defined by \Omega\\exp(\Omega=1 "Difference equation" redirects here It should not be confused with a Differential equation. to compute W. Together with the evaluation error estimate given in Chapeau-Blondeau and Monir, the following Python code implements this:

import math
 
def lambertW(x, prec = 1E-12, maxiters = 100):
    w = 0
    for i in range(maxiters):
        we = w * pow(math. Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability e,w)
        w1e = (w + 1) * pow(math. e,w)
        if prec > abs((x - we) / w1e):
            return w
        w -= (we - x) / (w1e - (w+2) * (we-x) / (2*w+2))
    raise ValueError("W doesn't converge fast enough for abs(z) = %f" % abs(x))

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 = ln(x + 1. 0);
              return 0. 665 * (1 + 0. 0195 * lx1) * lx1 + 0. 04;
      }
      return ln(x - 4. 0) - (1. 0 - 1. 0/ln(x)) * ln(ln(x));
}

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

History

Lambert first considered the related Lamberts Transcendental Equation in 1758 which lead to a paper by Leonhard Euler in 1783 which discussed the special case of we^w. Year 1758 ( MDCCLVIII) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common Year 1783 ( MDCCLXXXIII) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or However the inverse of we^w was first described by Pólya and Szegö in 1925. Year 1925 ( MCMXXV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian calendar.

References and external links

MathWorld - Lambert W-Function
Lambert function from Wolfram's function site.
Computing the Lambert Wfunction
Corless et al. Notes about Lambert W research
Corless et al. "On the Lambert W function" Adv. Computational Maths. 5, 329 - 359 (1996) (PDF)
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
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.

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