TR-00-20.html TR-00-20.mws ● Introduction ❍ History ❍ Lambert's Trinomial Equation (in Euler's format) ● Basic Facts about W ❍ Applications ■ Iteration of the exponential function (Euler, Carlsson, Hayes, Wright, ...) ❍ Complex Analysis ■ Branches ■ The Quadratrix of Hippias ■ An Animation of a few values of W(k,z) for varying z and k (recondite, says David) ■ Riemann Surface ■ The Animated Riemann Surface for The Lambert W function ■ A proof that the computed surface is correct ❍ Calculus ■ Derivatives ■ Series ■ Integration ❍ Computation ● A surprising integral ● The Wright omega; function ❍ Applications ❍ Complex Analysis ■ Branches ■ Riemann Surface ■ Actual plot of the Riemann Surface for omega;. ■ Proof that this computation works (incomplete) ❍ Calculus ■ Derivatives ■ Series ■ Integration ❍ Computation ● Concluding Remarks Still more fun results on the Lambert W function http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (1 of 41)9/14/2004 5:37:46 AM TR-00-20.html Robert M. Corless and David J. Jeffrey Department of Applied Mathematics University of Western Ontario Introduction This worksheet explores some recent results related to the function W(x), which satisfies > restart; > W(x)*exp(W(x)) = x; In fact, Maple knows this function rather well, and names it LambertW, following the paper ``On the Lambert W function'', by Corless, Gonnet, Hare, Jeffrey, and Knuth (Adv. Comp. Math. 1996); to save typing its real name all the time, we use > alias( W = LambertW ); This means now that any instance of W that occurs in this worksheet will use the short notation W instead of the long notation LambertW. > solve( y*exp(y) - z, y ); > _EnvAllSolutions := true; > solve( y*exp(y) - z, y ); > http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (2 of 41)9/14/2004 5:37:46 AM TR-00-20.html The usual reaction to this answer is another question: what on earth is W(x)? In this worksheet we will see some parts of the answer, and some of the history and applications of this function. The goal of this exposition is that, at the end, you feel comfortable with W(x) and are happy with it as an answer. Of course, to get to that point you will have to do some work, but luckily it is all rather pleasant. History The history of the function goes back to J. H. Lambert (1728-1777). This worksheet is not the right medium to discuss the life of Lambert, but a short note is appropriate. Lambert was born the son of a tailor, and was expected by his father to continue in that profession. His early fight for his education is a remarkable story---he had to sell drawings and writings to his classmates to buy candles for night study, for example---but eventually his talents were recognized and he got a position as tutor in a house which had a decent library. He then was able to educate himself, and went on to produce fundamental discoveries in cartography (the Lambert projection is still in use), hygrometry, pyrometry, statistics, philosophy (where he is actually more famous than as a mathematician), and pure mathematics. He is most noted as being the first person to prove that is irrational, which was an important step in proving that the classical problem of squaring the circle was impossible by straightedge and compass. Lambert's Trinomial Equation (in Euler's format) Consider the `trinomial equation' (using Euler's formulation, not Lambert's) > restart; > Euler_trinomial := x^alpha - x^beta = (alpha-beta)*v*x^(alpha+beta); > isolate( Euler_trinomial, v); > simplify(%,symbolic); http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (3 of 41)9/14/2004 5:37:46 AM TR-00-20.html > map( series, %, x=1, 6): > map(factor,%); Now reverse that series, to get a series for x in terms of v. > solve(%, x): > map(factor, %); > Lambert found that series as a solution to the trinomial equation, by hand calculation of the first few terms (his proof of the general case was, I believe, lost). His calculation was essentially similar to the above, except he did not carry so many terms, and used a more clever approach than the general series reversion techniques used by `solve' above; indeed his techniques proved a forerunner to the Lagrange Inversion Theorem which is now the classical way for reversion of series. Lambert at about the same time discovered that the series for x raised to an arbitrary power n had a very similar form: http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (4 of 41)9/14/2004 5:37:46 AM TR-00-20.html > x := %: > series( x^n, v, 5): > map(factor,%); > Lambert thought the pattern beautiful, and so simple that it must have a finite formula as a closed form. (I don't know of any, though, and I rather expect that it is in fact impossible). Euler came somewhat closer to proving that this simple pattern for the coefficients continues for all terms of the series; however, our translation of his Latin paper is as yet incomplete (Matt Davison has done most of it so far, and a student will complete the process this summer). Euler at least showed that the function > phi(alpha, beta, n, v); defined by the infinite series, satisfies > phi(alpha, beta, -alpha, v) - phi(alpha, beta, -beta, v) = (alpha-beta)*v; This does not mean that > phi(alpha,beta,n,v) = phi(alpha,beta,1,v)^n; http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (5 of 41)9/14/2004 5:37:46 AM TR-00-20.html > and this gap remained (unnoticed? if indeed Euler didn't close it himself in some part of the paper as yet untranslated). I do not know who first established that this series really does continue in the same way, but at the very latest, G. Raney in 1959 provided a combinatorial proof equivalent to the above identity (see Graham, Knuth, and Patashnik, Chapter 7). The series converges absolutely for > abs(v) < 1/(exp(1)*(sqrt( (alpha^2+beta^2)/2 ))); > Basic Facts about W Applications There are an extraordinary number of applications of W. This is not surprising, in retrospect: w*exp(w)=z is about the simplest transcendental equation that there is, that doesn't already have a solution in terms of known functions. A brief list of applications follows: Stability analysis for delay differential equations; counting rooted, labelled trees; mathematical models of dozens of unrelated phenomena, including water movement in soil, Wien's displacement law (discussed in the new paper by S.R. Valluri, D.J. Jeffrey, and R.M. Corless, ``S ome Applications of the Lambert W function to Physics ''), combustion, epidemics, jet fuel consumption; and education. Iteration of the exponential function (Euler, Carlsson, Hayes, Wright, ...) The following problem is sometimes posed on math contests, and is answered on the FAQ for sci.math (or was, for a while). > x^(x^(x^(`.`^(`.`^(`.`))))) = 2; http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (6 of 41)9/14/2004 5:37:46 AM TR-00-20.html What is x? Once you've solved that one, now find x such that the left hand side is equal to 4. More generally, if we replace 2 by y, what is x as a function of y? What is y as a function of x? We are quickly led to the equation > x^y = y; This can be easily solved, in Maple: > solve(%,y); This tells us the limiting value of that iteration, when it converges. For more information, see the paper by Baker and Rippon in the Monthly from 1982. Further, this iteration does not converge for all complex x. For some x, indeed, it settles to a two-cycle. For some, it settles to a three-cycle. Hayes first established that there were complex x-values giving rise to cycles in this iteration of arbitrary length. Drawing those x-values, in the right coordinate system, leads to a lovely image, where the regions of cycles are bordered by what looks like a fractal. Complex Analysis Branches The Quadratrix of Hippias We choose the branch cuts for W(k,z) by looking at the image of the negative real axis: > restart; > w := u + I*v; http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (7 of 41)9/14/2004 5:37:46 AM TR-00-20.html > z := evalc(w*exp(w)); > x := evalc(Re(z)); > y := evalc(Im(z)); > solve(y=0,u); > plot({seq([-v*cos(v)/sin(v),v,v=(k-1)*Pi*(1+(k-0.01)/1000)..k*Pi*(1-k/1000)],k=-3..4)}, view=[-4*Pi..4*Pi,-4*Pi..4*Pi],colour=black); http://www.mapleapps.com/categories/mathematics/algebra/html/TR-00-20.html (8 of 41)9/14/2004 5:37:46 AM TR-00-20.html That is a subset of the curve known since antiquity as the Quadratrix of Hippias. The first curve ever named after its discoverer, Hippias of Elis (ca. 400BC), it can be used to square the circle, trisect the angle and even duplicate the cube. It is, therefore, not constructible by straightedge and compass. But we only need those parts of it that satisfy x < 0. So we must look at the sign of x. > X := simplify(subs(u=-v*cot(v),x)); > plot( X, v=-4*Pi..4*Pi, view=[-8..8,-2..2], numpoints=101, discont=true ); That allows us to deduce which pieces of the curve demark branch cuts for the Lambert W function.