Another Incarnation of the Lambert W Function
aa r X i v : . [ m a t h . C A ] M a r ANOTHER INCARNATION OF THE LAMBERT W FUNCTION
ALEXANDER KHEYFITS
To the memory of Jonathan M. Borwein (1951-2016)July 23, 2018
Abstract:
The Lambert W function was introduced by Euler in 1779, but wasnot well-known until it was implemented in Maple, and the seminal paper [1] waspublished in 1996. In this note we describe a simple problem, which can be straight-forwardly solved in terms of the W function. At the recent seminar, Professor Bertram Kabak asked the following question:
The graphs of the exponential function e x and of its inverse, the natural logarithm ln x have no point in common, see Fig. 1. Consider an exponential function with abase b, < b, b = 1 , y = b x . For what values of the base b , have the graphs of this function and its inverse, y ( x ) =log b x the common points, and how many? − − − xy y = e x y = ln x Figure 1.
Graphs of exponential function y = e x and its inverselogarithmic function y = ln x never intersect.It turns out that this problem can be easily solved by making use of the Lambert W function W = w ( z ). The latter is given implicitly for every complex number z by the equation(1) we w = z, where we follow the standard-now notation w = w ( z ) [1].1234 − − − − wz z = we w − z = − /ez = 2 . z = − . W W − ( − , − /e ) Figure 2.
Function z = we w . The dashed lines are the horizontaltangent line z = − /e and the horizontal lines z = 2 . z = − . W was implemented in Maple and other major CASs, and the article [1]has been published, the number of publications devoted to the function, has growndramatically . The function enjoys so many applications, that together with thelogarithmic function, the W should be in a toolbox of any researcher. Some authorsclaim that the W is an elementary function. Leave it to the individual judgment,whether and in what sense the Lambert W function is elementary, we show how the W naturally occurs in an elementary problem.The Lambert W is a many-valued analytic function, therefore, its complete studyand, in particular, description of its single-valued branches, should be done in complex-analytic terms, which is beyond the scope of this brief note, see, e.g., [1] or [3], wherethe closed-form representation of all the branches of W in terms of contour integralswas derived.We mention only few necessary properties of the W function, referring the readerto the papers above. Begin by graphing the left-hand side of (1), see Fig. 2. Thefunction f ( w ) = we w has the global minimum − /e at w = −
1. For every z ≥ z = we w has exactly one real root, that is, the Lambert W function has one On Jan. 8, 2017, the query ’Lambert W function’ returned 302 references on MathSciNet, about450,000 references on Google, and more than 1,200,000 references on Google Scholar. NOTHER INCARNATION OF THE LAMBERT W FUNCTION 3 real-valued branch. If − /e < z <
0, equation (1) has two real roots, − < W < W − < −
1. When z = − /e , these two roots merge in the double root w = − z < − /e , these roots disappear from the graph, because they became complexnumbers.The function f ( w ) = we w − z is an entire function for every complex z ; suchfunctions are called quasi-polynomials. It is distinct from e w , therefore, for any com-plex z = 0 this function has infinitely many roots. Since every root generates its ownsingle-valued branch , the inverse function, Lambert W has infinitely many branches,and only two of them are real-valued on the real axis. They are conventionally calledthe principal branch, W ( z ), which is real for z ≥ − /e , and the branch W − ( z ),real-valued for − /e ≤ z < W ( z ) is the mirror reflection, with respect to the bisectrix w = z ,of the right half, w ≥ − , z ≥ − /e , of the graph in Fig. 2. The graph of W − ( z )is the mirror reflection of the left half of the same graph, w ≤ − , − /e ≤ z ≤ z = w .246 − − xyy = log . xy = (0 . x Figure 3.
The functions y = (0 . x and y = log . x have the uniqueintersection point. The dashed line is the bisectrix y = x .Now we can take up the question above. First, let be 0 < b <
1. The graphs ofa function and its inverse are symmetrical with respect to the bisectrix z = w , inthe notations of Fig. 2, or the bisectrix y = x in the notations of Fig. 3. Therefore,the intersection points, if any, must belong to the bisectrix, that is, satisfy theequation b w = w , or ( − ln b ) we − (ln b ) w = − ln b . The right hand side here is positivebecause b <
1, hence from Fig. 2 it follows that for any 0 < b < b w andlog b w . It is worth repeating that the intersection point is given by the principalbranch W (ln(1 /b ) of the Lambert W function. For instance, the case b = 0 . ALEXANDER KHEYFITS W ( − ln 0 . / ( − ln 0 . ≈ .
83. If we depart from the equation w = log b w , we arriveat the same conclusion.The case b > x = b x , or to equation (1), we w = z, where we are to set w = − x ln b and z = − ln b <
0. Since now b > x > w = − x ln b <
0. Therefore, if − /e < z = − ln b <
0, that is, if 1 < b < e /e ,then there are two intersections, W ( z ) and W − ( z ), given by the principal branch W and the preceding branch W − of the W function, see Fig. 4.If − /e = z , where z = − ln b , that is, b = e /e , then the two points merge intothe point of tangency, see Fig. 5. We can state the conclusion as follows.2468 2 4 6 8 10 12 − xy y = log b xy = b x Figure 4.
The functions y = b x and y = log b x with b = 1 . Proposition 1.
For < b < , the graph of the exponential function y = b x andthe graph of its inverse y = log b x have the unique point of intersection, given by theprincipal branch of the Lambert W function W ( − ln b ) / ( − ln b ) . The graphs of theexponential function y = b x and of its inverse y = log b x have a point of tangency ifand only if b = e /e . The tangency point is x t with − x t ln b = − , that is, x t = e .These graphs intersect exactly twice if and only if < b < e /e . The two pointsof intersection are given by the two branches, W ( − / ln b ) and W − ( − / ln b ) . Forexample, if b = 1 . , the intersection points are (Fig.4) − W ( − / ln 1 . / ln 1 . ≈ . / ln 1 . ≈ . NOTHER INCARNATION OF THE LAMBERT W FUNCTION 5 and − W − ( − / ln 1 . ≈ . / ln 1 . ≈ . . For b > e /e , these graphs do not intersect (Fig. 1). (cid:3) − − xy O y = log b xy = b x The common point ( e, e ) Figure 5.
The functions y = b x and y = log b x with b = e /e ≈ . Problem.
Describe those continuous increasing functions f ( x ) , < x < ∞ ,whose graphs have a) one, b) two, c) none, d) n ≥ points of intersection with theinverse function f − . The additional assumption of convexity will, definitely, sim-plify the problem and likely, lead to the functions e f ( x ) . Acknowledgement.
The author is thankful to Professor German Kalugin foruseful remarks, which help to improve the manuscript.
References [1] Corless, R. M., G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth. On the LambertW function,
Advances in Computational Mathematics , Vol. 5(1996) 329-359.[2] Hayes, B. Why W?