Applied Mathematics, 2013, 4, 887-892 http://dx.doi.org/10.4236/am.2013.46122 Published Online June 2013 (http://www.scirp.org/journal/am) A Brief Look into the Lambert W Function Thomas P. Dence Ashland University, Ashland, USA Email: [email protected] Received December 4, 2012; revised March 14, 2013; accepted March 22, 2013 Copyright © 2013 Thomas P. Dence. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The Lambert W function has its origin traced back 250 years, but it’s just been in the past several decades when some of the real usefulness of the function has been brought to the attention of the scientific community. Keywords: Lagrange Inversion Theorem; Infinite Tower of Exponents 1. Introduction defined on the entire real line, and each being one-to-one; but the product is not injective. Consequently, if we re- The Lambert W function, named after Johann Heinrich strict the domain to x 1, then xex will possess an Lambert [1], is a standard function in both Mathematica, inverse, which is a function, and it’s this function that is where it’s called Product Log x , and in Maple, where now known as the (principal) Lambert W function (Fig- you can use both Lambert W x or Lambert W0, x . ure 1(b)), written as W x . An alternative branch for The zero in this latter expression denotes the principal x x W would be defined for that portion of xe when branch of the inverse of xe . The actual usage of the letter W has a rather vague origin. One source attributes x 1. We won’t consider that situation in this article. it to some earlier papers on the subject that wrote the Several function values of W are easy to compute, wxew 1 standard equation as using a small w. Pro- such as W1 e 1,W00 . Wln2ln2, gramming protocol with Maple then forced the letter to 2 be capitalized [2]. Another source [3] attributes the W to and We 1. The value of W1 , known as the honor the British mathematician Sir Edward M. Wright omega constant, has the approximate value 0.567143. (famous co-author with G. H. Hardy of An Introduction The number W1 is, in some sense, a distant cousin of to the Theory of Numbers) who did a lot of pioneering the golden ratio , since 1 is a solution to 11x x , work with the function. Finally, Robert Corless and and W1 is the solution to 1ex x , and x 1 is the David Jeffrey of the University of Western Ontario have linear Maclaurin approximation to ex (Figure 2). Since written, during the past several decades, a number of W is the inverse of xex , it follows that W1e W11 journal articles on the function. Their paper in 1996, in and that the slope of the curve in Figure 1(b) at the point 1 collaboration with Gaston Gonnet, David Hare, and Don- W1 1, W 1 is 1 e 0.6381. ald Knuth, was where Lambert’s name got attached to the function [2]. It could have been coined the Euler W function, since Euler had studied the equation xex 3. Computation [4] (although Euler credits Lambert as studying the equa- A natural question is how to compute arbitrary values of tion first [5]), but they decided Euler had enough items W x . One result, from the Lagrange inversion theorem, attached to his name! asserts that the Lambert W function has the Taylor series expansion [6,7] 2. Definition n1 x n The exponential function y xe is defined for all real Wx xn , (1) 1 n1 n! x, but has a codomain of y . This function (Figure e which, unfortunately, has a radius of convergence of 1(a)) is the product of two elementary functions, each merely 1 e . Since the denominator n! grows rapidly it’s Copyright © 2013 SciRes. AM 888 T. P. DENCE advantageous to write the series with the coefficients n defined recursively as W x cxn , with n1 n2 n cc11, nn cn1, 2 . This recursion lends n 1 itself to easy programming evaluation. Testing this, with say a series of 150 terms (which is plenty, considering 1 1 that x ), with x ln 2 0.3465736 , we ob- e 2 tain a partial sum value of S 0.69314684 , which dif- fers from the exact value of ln 2 by 0.0000003. We 1 also note that x , so the use of the series is justified. e On the other hand, a TI-graphing calculator returns “overflow error” if we try to determine W1, primarily (a) since the coefficients grow rapidly. 1 Suppose that x and we wish to compute W x . e One possibility is the series n 1 Wlnlnlnx xx n an x (2) n1 ln x m n m ln ln x where axn 1S, nnm 1 and m1 m! S,nk denotes a Stirling number of the first kind [3]. The series (2) is somewhat impractical to use because of (b) the difficulty in determining axn ; it turns out to be Figure 1. (a) Graph of y xe x ; (b) Graph of W(x), more useful to employ some standard numerical schemes 1 for approximating W x . x . y e First, setting W x y , we need to solve ye x . y Defining the function g by g yye x, we use Newton’s method to approximate y in gy 0 . This 2 yn gy n yxn e gives yynn1 . To determine gynny1 W2 , for example, starting with an initial approximate of y0 0 , after 7 more iterations we get y8 0.852605502 , which is an excellent approximation y8 to W2 because y8e returns 2 on the calculator. If x is a relatively small number, then an initial approximate of 0 will suffice for the algorithm; but if x is large, then ln x can be chosen for y0 . For instance, if x 10 , choose y0 ln 10 , and after 5 iterations we get W(10) ≈ 1.745528003. Newton’s method is a favorite iteration scheme for many because of its simplicity, though the convergence, quadratic in general, is typically relatively slow. A faster choice is furnished by Halley’s method (of Halley’s comet fame), which produces cubic convergence, and happens to be the choice implemented by the software Figure 2. W1 and 1 . Maple; this scheme gives [8] Copyright © 2013 SciRes. AM T. P. DENCE 889 yn 2W yxne eW2 yynn1 . Rewriting W as puts this into yyx2eyn 3 y nn 1W y 1en n n 22yn the form which fits the general case for W [5]. In Employing this gives W(10) ≈ 1.745528003 after 3 it- fact, from this form, we readily see that there is a point of erations. This complex looking scheme is actually what inflection on the curve when W2, which actually you get when you apply Newton’s method to the function falls on the other branch of the W function. g x Continuing along the calculus vein, we should exam- [9]. An alternative root-finding scheme, using ine, if possible, the integral of W x . To this end, recall gx that y W x iff ye y x . Thus, continued fraction expansion, is described in [10]. ddyyy Wdx xyx 1W x WW xx 4. Calculus x yy2 We know that since y xe is an increasing and dif- yyeed y ferentiable function for all x 1 then its inverse y W x is likewise increasing and differentiable for and integrating this last integral by parts, we obtain 1 1 all x . xy 1 , which now gives e y Differentiating this latter equation with respect to y, 1 we obtain WxdW1 x x x. C (4) W x dW dx dW 1ey yy e, ddxy d x In particular, the area of the region bounded by the curve W x , the x-axis, and the line x e is, there- so fore, 1 y e W x e yy 1 yyy1e 1 e Wdxxx W x 1 (3) Wx 0 W x y W x e 0 ,0x . yx11WW1xx x 1 ee111 limW 1 1. 11 0 W In particular, We , and similarly, ee11 2 We note this result agrees with evaluating the integral lim Wx . What about W0 the right-hand via inverse functions [11], because then xe1 e Wx side of (3) is indeterminant at x 0 , but division of both Wdxxe W e 0W0 x ed1x xe . sides of (1) by x and taking the limit as x 0 give 00 Wx lim 1 . This yields Other integrals, involving functions containing W, can x0 x be computed, some just with a special change of variable WWxxx1 [6]. For instance, W0 lim lim . 111 xx00xx1W 1W x 1 2Wx 2 x Wdxx e Wx W x C. 222 For large x, the graph of W x bears strong resem- The function xW x is concave up, connecting 0,0 W x e blance to ln x , since from (2) we have lim 1 , 2 x ln x and ee, , hence its area xxxeWd 318 0 although we have to be careful here because the difference e Wlnx x increases without bound as x [7]. is less than xxd e2 2. Similarly we find The graph of W x , like that of ln x , is concave down- 0 x ward for all x since xe is concave upward.