{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 3 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 13 " Still more " }{TEXT 267 3 "fun" }{TEXT -1 24 " results on\nthe Lambert " }{TEXT 256 1 "W" }{TEXT -1 10 " function " }}{PARA 19 "" 0 "" {TEXT -1 102 "Robert M. C orless and David J. Jeffrey\nDepartment of Applied Mathematics\nUniver sity of Western Ontario" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "This worksheet explores some recent results related to th e function W(x), which satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "W(x)*exp(W(x)) = x;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "In fact, Maple knows this function rather well, and names it L ambertW, following the paper ``On the Lambert " }{TEXT 257 1 "W" } {TEXT -1 132 " function'', by Corless, Gonnet, Hare, Jeffrey, and Knut h (Adv. Comp. Math. 1996); to save typing its real name all the time, \+ we use" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "alias( W = Lamber tW );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "This means now that any instance of W that occurs in this worksheet will use the short notati on W instead of the long notation LambertW." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve( y*exp(y) - z, y );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 78 "The usual react ion to this answer is another question: what on earth is W(x)?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 334 "In this \+ worksheet we will see some parts of the answer, and some of the histor y and applications of this function. The goal of this exposition is t hat, 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 som e work, but luckily it is all rather pleasant." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "History" }}{PARA 0 "" 0 "" {TEXT -1 172 "The history of the function goes back to J. H. Lambert (1728-1777). This worksheet is not the right medium to discu ss the life of Lambert, but a short note is appropriate." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 703 "Lambert was born \+ the son of a tailor, and was expected by his father to continue in tha t profession. His early fight for his education is a remarkable story ---he had to sell drawings and writings to his classmates to buy candl es for night study, for example---but eventually his talents were reco gnized and he got a position as tutor in a house which had a decent li brary. He then was able to educate himself, and went on to produce f undamental discoveries in cartography (the Lambert projection is still in use), hygrometry, pyrometry, statistics, philosophy (where he is a ctually more famous than as a mathematician), and pure mathematics. H e is most noted as being the first person to prove that " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 148 " is irrational, which was an impor tant step in proving that the classical problem of squaring the circle was impossible by straightedge and compass." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Lambert's Trinomial Equation (in Euler's format)" }}{PARA 0 "" 0 " " {TEXT -1 76 "Consider the `trinomial equation' (using Euler's formul ation, not Lambert's)" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "Euler_trinomial := x^alpha - x^beta = (alpha-beta)* v*x^(alpha+beta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "isolat e( Euler_trinomial, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " simplify(%,symbolic);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ma p( series, %, x=1, 5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "m ap(factor,%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Now reverse that series, to get a series for x in terms of v." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(%, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "map(factor, %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 490 "Lambert found that series as a solution to the trinomial equation, by hand calculation of the first few terms (his proof of the general cas e was, I believe, lost). His calculation was essentially similar to t he above, except he did not carry so many terms, and used a more cleve r approach than the general series reversion techniques used by `solve ' above; indeed his techniques proved a forerunner to the Lagrange Inv ersion Theorem which is now the classical way for reversion of series ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Lam bert at about the same time discovered that the series for x raised to an arbitrary power n had a very similar form:" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x := %:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series( x^n, v, 5):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "map(factor,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "Lambert thought the pattern be autiful, 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)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 324 "Euler came somewhat closer to proving that this simple p attern for the coefficients continues for all terms of the series; how ever, our translation of his Latin paper is as yet incomplete (Matt Da vison has done most of it so far, and a student will complete the proc ess this summer). Euler at least showed that the function " }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "phi(alpha , beta, n, v);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "defined by the \+ infinite series, satisfies" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "phi(alpha, beta, -alpha, v) - phi(alpha, \+ beta, -beta, v) = (alpha-beta)*v;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "This does not mean that " }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "phi(alpha,beta,n,v) = phi(alpha,bet a,1,v)^n;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 365 "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 c ontinue in the same way, but at the very latest, G. Raney in 1959 prov ided a combinatorial proof equivalent to the above identity (see Graha m, Knuth, and Patashnik, Chapter 7)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The series converges absolutely for " } }{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "abs(v) < 1/(exp(1)*(sqrt( (alpha^2+beta^2)/2 )));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" } }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Basic Facts about W" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Applications" }}{PARA 0 "" 0 "" {TEXT -1 272 "There are an extraordinary number of applications of W. This \+ is not surprising, in retrospect: w*exp(w)=z is about the simplest tra nscendental equation that there is, that doesn't already have a soluti on in terms of known functions. A brief list of applications follows: " }}{PARA 0 "" 0 "" {TEXT -1 278 "Stability analysis for delay differe ntial equations; counting rooted, labelled trees; mathematical models \+ of dozens of unrelated phenomena, including water movement in soil, Wi en's displacement law (discussed in the new paper by S.R. Valluri, D.J . Jeffrey, and R.M. Corless, ``S" }{TEXT 261 53 "ome Applications of t he Lambert W function to Physics" }{TEXT -1 64 "''), combustion, epide mics, jet fuel consumption; and education." }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 16 "Complex Analysis" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Branches" }}{PARA 0 "" 0 "" {TEXT -1 167 "Now let's look at the compl ex branches of W. There is a symmetry relation amongst the branches, b ut it doesn't hold on the branch cuts unless we use a `signed zero'. \+ " }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "restart;\nalias(W=LambertW):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fewvals := a -> [seq(evalf(W(k,a)),k=-6..5)];" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fewvals(-0.2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "pts := a -> map( t->[evalc(Re(t)),e valc(Im(t))], fewvals(a));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "N := 50; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "p := array(1..N);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "avals := [seq( evalf(-Pi/2*k /N),k=1..N)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for i to N do p[i] := plot(pts(avals[i]),style=POINT,symbol=DIAMOND,symbolsize=3 0); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plots[display]([ seq(p[i],i=1..N)],insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 16 "Riemann Sur face\n" }}{PARA 0 "" 0 "" {TEXT -1 308 "The following shows an animati on of the Riemann Surface for the Lambert W function. Strictly speaki ng, we have to prove that the technique works, by verifying that given (x,y,v) we can find u; that is, there is a bijection between a point \+ on the 3d surface and the mapping z -> W(z) (i.e. (x,y) -> (u,v) ). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 55 "The Animated Riemann Surface for The Lambert W function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "w := u+I*v; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "z := evalc(w*exp(w)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x := evalc(Re(z)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y := evalc(Im(z)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "B := 6: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "f := theta -> pl ot3d([x,y,v], u=-6..1, v=-B..B, grid=[50,50], orientation=[theta,80], \+ color=u): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(plots): \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "bunch := [seq(f(10*k),k =-17..18)]: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "display(bunch, insequence=true, style=PAT CHNOGRID, axes=NONE, view=[-1..1,-1..1,-B..B]);\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 20 "" 0 "" {TEXT -1 44 "A proof that the computed surfa ce is correct" }}{PARA 0 "" 0 "" {TEXT -1 46 "We must show that given \+ (x,y,v) we can find u." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "w := u+I*v; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "z := evalc(w*exp(w)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "x := evalc(Re(z)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y := evalc(Im(z)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " X/Y = normal( x/y );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sol ve( %, u );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "That solution is \+ unique. Now we must investigate the condition y=0, which would have p revented us doing the division X/Y above." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "y=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "One sol ution is v=0 (independently of u). If v=0, then u is determined by th e x equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "X = eval( \+ x, v=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "There is a unique re al solution if X >= 0, and two real solutions if -1/e <= X < 0, and no real solutions otherwise (this just follows from a graph)." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Now, we should see if there are a ny possible solutions if v is not zero. If sin(v)=0, but v is not zer o, then cos(v) = (-1)^k." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "0 = eval(y, \{sin(v)=0, cos(v)=(-1)^k\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "This can happen only if u is -infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Therefore our pictu re is good except on the line -1/e <= z < 0, when we have a spurious i ntersection." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Calculus" }}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 11 "Derivatives" }}{PARA 0 "" 0 "" {TEXT -1 23 "Differentia tion of W(x)" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "restart; \nalias(W=LambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(W(x),x);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "normal(diff(W(x),x$5));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 18 "diff(W(exp(x)),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "normal(diff(W(exp(x)),x$5));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot( \{W(exp(x)),W(x)\}, x =-1..2 );" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Series\n" }}{PARA 0 "" 0 "" {TEXT -1 29 "Now for some fun with series." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "restart; \nalias(W=Lambe rtW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "series(W(x), x, 8) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "The general term there is ( -n)^(n-1)/n!. This is the exponential generating function for unroote d, labelled trees (apart from sign)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "add( (-n)^(n-1)/n!*x^n, n=1..7 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Now a branch point series:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "series(W(x),x=-1/exp(1));" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "Let's get rid of those ugly square roots and exp(- 1)'s:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "series(W((p^2/2-1) *exp(-1)), p);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Maple doesn't \+ know the asymptotic series for W, at least not directly (because, to ` asympt', W is an answer, not a question...)" }}{PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "asympt(W(x),x);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "But we can trick Maple into giving us what we want:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "alias( omega = RootOf( y + l n(y) - z , y ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "map( f actor, asympt( omega, z, 5 ) );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "We will see later that this is an interesting function itself; for now," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve( y + ln(y) = z, y );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Therefore, omega(z) \+ = W(exp(z)), and so the asymptotics for W can be determined in Maple f rom the asymptotics of omega." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Integration\n" }}{PARA 0 "" 0 "" {TEXT -1 254 "``Everybod y knows'' that W(z) is not elementary. If it's true, where's the proo f? Well, it hasn't been done, yet! James Davenport is currently worki ng with David and me on completing the proof that W is not expressible in terms of elementary functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 156 "Nonetheless, we may integrate some expre ssions containing the Lambert W function, by the simple change of vari able w = W(x) (and hence dx = (w+1)exp(w)dw )." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "restart; \nalias(W=LambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "int( sin(W(x)),x);" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "plot(%,x=0.. 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F := int( W(x)*ln(W( x)), x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eval( F, x=2.3 \+ );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot( \{evalc(Re(F)), evalc(Im(F))\}, x=-1/10..1 , colour=black);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Computa tion\n" }}{PARA 0 "" 0 "" {TEXT -1 234 "The method of choice in Maple \+ to compute numerical values of W is Halley's method, which is a third- order variant of Newton's method. In fact, arbitrary-order methods fo r the computation of W exist, and are continually rediscovered; " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "f := w -> w*exp(w) - z;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 64 "Halley := w -> w - f(w)/(D(f)(w) - 1/2*D(D(f ))(w)*f(w)/D(f)(w));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "z : = rand()/Float(1,12);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "computed_w[0] := z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "residual := computed_w[0]*exp(compu ted_w[0]) - z;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Digits := 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 81 "for k to 4 do Digits := 3*Digits; computed_w [k] := Halley( computed_w[k-1] ); od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Digits := Digits + 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "residual := computed_w[4]*exp(computed_w[4]) - z;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 "A surprising integral" }}{PARA 0 "" 0 "" {TEXT -1 53 " Let us define the following strange-looking integral." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Define the integrand, which depends on a parameter z." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "f := ((1-v*cot(v))^2 + v^2 ) /(z + v*csc(v)*exp(-v*cot(v)));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Let's take a typical value of z, and plot the integrand:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot( eval(f,z=1), v=-Pi..Pi , -0.1..1.5 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F := Int( f, v=0..p )/Pi;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "From R. Willi am Gosper:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Finally, I must publicly \"lambaste\" Rob Corless for his prepo sterous allegation that\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "W(z)/z = subs(p=Pi, F);" }}}{PARA 0 "" 0 "" {TEXT -1 67 "\nSimply because Taylor expanding and numerical ly integrating gives " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "series( F, z ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map( a->evalf(subs(p=3.1,a)), % );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalf( series( LambertW(z), z ) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 569 "\n...is no reason to go about endangering public morals \+ and threatening massive unemployment via wild claims that implicit tr ascendental equations can be inverted with ordinary definite integrals . Sheesh!\n\nThink of the numericists. Think of their families! (Th e numerical integrator converged slambangularly on these. This integ ral may well provide new approximations.)\n\nRMC again: in fact, this \+ integral is not the first integral expression for W. There is another , by Siewert and Burniston, which dates to 1975. It is more complicat ed than the above, however.\n" }}{PARA 0 "" 0 "" {TEXT 260 7 "Theorem " }{TEXT -1 2 ": " }{TEXT 262 30 "W(z)/z is a Stieltjes function" } {TEXT -1 303 ", and hence its continued fraction has some well-underst ood properties (including the fact that all the poles of the subdiagon al Pade approximants of increasing degrees are real and interlace, for example). Proof: establish the integral above by use of the Cauchy i ntegral formula and a transformation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "alias(W=LambertW):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "series( W(z)/z, z, 8 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "convert( %, ratpoly );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve( deno m(%), z, complex );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "seri es( W(z)/z, z, 10 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "con vert(%, ratpoly):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve ( denom(%), z, complex );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Tim e permitting, we will explore the following formula for a bi-infinite \+ family of zeros of the denominator of the integrand f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "denom( f );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "zeros := (W(k,z)-W(m,z))/(2*I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs( v=zeros, convert( denom(f), e xp ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify( % );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Petr Lisonek helped with the ve rification of that particular discovery." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "The Wright " }{XPPEDIT 18 0 "omega;" "6#%&omegaG " }{TEXT -1 9 " function" }}{PARA 0 "" 0 "" {TEXT -1 11 "The Wright " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 19 " function satisfies " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "omega + ln(omega) = z;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "It is clearly a cognate of W:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "_EnvAllSolutions := true; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve( y + ln(y) = z, y );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 87 "However, there are reasons (as we will see) for a \+ separate existence for this function." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Applications" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 235 "There are several applications of this function; \+ indeed, some of them have been listed already as applications of W. B ut, because of superior numerical properties and other advantages, the y are probably best listed as applications of " }{XPPEDIT 18 0 "omega; " "6#%&omegaG" }{TEXT -1 98 " directly. The first to come to our atte ntion was the need in convex optimization. Consider the " }{TEXT 258 16 "convex conjugate" }{TEXT -1 10 ", namely, " }{XPPEDIT 18 0 "(f^`*` )(s) = sup[r](r*s-f(r));" "6#/-)%\"fG%\"*G6#%\"sG-&%$supG6#%\"rG6#,&*& F.\"\"\"F)F2F2-F&6#F.!\"\"" }{TEXT -1 19 ", of the function " } {XPPEDIT 18 0 "f(r) = r*ln(r/(1-r))-r;" "6#/-%\"fG6#%\"rG,&*&F'\"\"\"- %#lnG6#*&F'F*,&F*F*F'!\"\"F0F*F*F'F0" }{TEXT -1 47 ". Calculation show s that the conjugate is just " }{XPPEDIT 18 0 "W(exp(s));" "6#-%\"WG6# -%$expG6#%\"sG" }{TEXT -1 6 " , or " }{XPPEDIT 18 0 "omega(s);" "6#-%& omegaG6#%\"sG" }{TEXT -1 246 ". For large s, computing this function \+ meant first exponentiation (making a REALLY large number, with possibl e overflow difficulties) and then taking (essentially) its logarithm. \+ For some purposes, it is better to work directly with the Wright " } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 10 " function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 330 "%1. J.M. Borwe in and A.S. Lewis, , CMS Advanced Books in Mathematics, to appear.\n@b ook\{BorweinLewis:1999,\n author = \{Jonathan M.~Borwein and Adrian S .~Lewis\},\n title = \{Convex Analysis and Nonlinear Optimization\}, \n year = \{to appear\},\n publisher = \{Canadian Mathematical Socie ty\},\n series = \{Advanced Books in Mathematics\},\n\}" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 229 "Another applicati on is discussed in S.R. Valluri, D.J. Jeffrey, and R.M. Corless, ``Som e Applications of the Lambert W function to Physics'', namely the conf ormal mapping for the fringing fields of a capacitor. We need to sol ve" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "1 + z + exp(z) = zeta;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 " solve(%, z );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "This is better e xpressed (correctly accounting for branches, which Maple ignores above ) as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "z = sort( zeta - 1 \+ - omega( zeta - 1 ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Complex Analysis" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Branches" }}{PARA 0 "" 0 "" {TEXT -1 45 "P robably the best reason to even bother with " }{XPPEDIT 18 0 "omega" " 6#%&omegaG" }{TEXT -1 45 " is the fact that its branching behaviour is " }{TEXT 259 4 "much" }{TEXT -1 24 " simpler than that of W." }} {PARA 0 "" 0 "" {TEXT 263 7 "Theorem" }{TEXT -1 73 ": (Wright 1959, Si ewert & Burniston 1975, Jeffrey, Corless & Hare 1995): " }{TEXT 264 51 "The solution of the equation y + ln y = z is unique" }{TEXT -1 95 ", except if z = t + I*Pi for t <= -1, when there are exactly two solu tions, which we denote by " }{XPPEDIT 18 0 "omega" "6#%&omegaG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "omega[-1];" "6#&%&omegaG6#,$\"\"\"! \"\"" }{TEXT -1 263 ". We call the line z=t+I*Pi the ``doubling line' ', and its reflection t-I*Pi the ``reflected line''. The solution is \+ unique on the reflected line. The solutions are discontinuous on both the doubling line and its reflection, as we will see in the next sect ion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "doubling := plot( [t,Pi,t=-3..-1],v iew=[-3..1,-4..4], linestyle=1,colour=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "reflection := plot( [t,-Pi,t=-3..-1],view=[-3..1 ,-4..4],linestyle=3,colour=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plots[display](\{doubling,reflection\});" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 36 "Relations of the branches of W with " } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "Except on the doubling line, we have that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "W(k,z) = omega( ln(z) + 2*Pi*I*k ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "omega(z) = W(unwindK(z) ,exp(z));" }}}{PARA 0 "" 0 "" {TEXT -1 48 "On the doubling line, we ha ve, for k=0 or k=-1, " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ome ga[k](z) = W(k,exp(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " W(k,z) = omega[k](ln(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "Riemann Surface" }}{PARA 0 "" 0 "" {TEXT -1 35 "We examine the Riemann Surface for " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 49 " in the same manner that we l ooked at that for W." }}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 39 "Actual p lot of the Riemann Surface for " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z := x+I*y; omega := mu + \+ I*nu;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x := evalc(Re(omeg a+ln(omega)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y := eval c(Im(omega+ln(omega)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "plot3d( [x,y,mu], mu=-4..2, nu=-4..4, colour=black, axes=BOXED, style =PATCHNOGRID, labels=[\"x\",\"y\",\"mu\"], view=[-2..1, -5..5, -5..3], grid=[200,200], style=POINT );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 46 "Proof that this computation works (incomplete)" }}{PARA 0 "" 0 "" {TEXT -1 223 "We must check that given 3 of the coordinates, x, y, and mu as above, we can uniquely identify the nu, the fourth co ordinate. If we can, then there is a bijection between the points on \+ the 3-d surface and the mapping z to " }{XPPEDIT 18 0 "omega;" "6#%&om egaG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta rt:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z := x+I*y; omega := mu + I*nu;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "x := evalc(R e(omega+ln(omega)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "y : = evalc(Im(omega+ln(omega)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve( x=X, nu );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 270 "There are apparently two values of nu. However, if y and nu are not zero, \+ then the equation with the arctan above uniquely determines the sign o f nu, because y cannot be nu + arctan(nu,mu) and -nu + arctan(-nu,mu) \+ simultaneously. If y is not zero, but nu is zero, then" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "y = 0 + arctan(0,mu);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "This means that the arctan must be Pi, a nd mu must be less than zero. Therefore the x-equation becomes" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "_EnvAllSolutions := true;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "X = mu + ln(-mu);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(%,mu);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 187 "Therefore, if nu=0 (which is a double ro ot), there are two values of mu corresponding, but this is still bijec tive because it is (x,y,mu) that are fixed, and nu that we have to det ermine." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Now it only remains to examine the case y=0 but nu not zero. " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "last_case := eval([X=x,0=y] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 254 "If there are any zeros of \+ this at all, then there are two values of nu corresponding to every va lue of mu, because these equations are symmetric under the change of s ign nu -> -nu; this will produce a double root at nu=0, of course, but this is excluded." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "possibilities := solve( last _case[1], nu );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "subs( nu =possibilities[1], last_case[2] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Ignoring branches, we have" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "tan(sqrt(-mu^2+exp(X-mu)^2)) = - sqrt(-mu^2+exp(X-mu) ^2)/mu;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 173 "At this point I give \+ up for today! We can appeal to the uniqueness theorem (we have found \+ a solution for nu=0 already) but we ought to be able to verify this in dependently." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 20 "" 0 "" {TEXT -1 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Calculus" }}{SECT 1 {PARA 5 " " 0 "" {TEXT -1 11 "Derivatives" }}{PARA 0 "" 0 "" {TEXT -1 18 "The de rivative of " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 36 " is s lightly simpler than that of W." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "restart; \nalias(omega=RootOf(y+ln(y)=z,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "normal( diff( omega, z ) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "normal( diff( omega, z$5 ) );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 6 "Series" }}{PARA 0 "" 0 "" {TEXT -1 11 "Series for " } {XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 22 " are likewise simple r." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series( omega, z=1 ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "alias( w=LambertW(exp(a )) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series( omega, z=a );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "series( omega, z=-1+ I*Pi );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "So, Maple doesn't kno w how to do that series in this form. In fact, that series is complet ely known, and is related to Stirling's formula for n! (see Corless, J effrey, and Knuth 1997)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "alias(W=LambertW);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seri es(W(-exp(-1-p^2/2)),p,10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "asympt(GAMMA(n),n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 11 "Integration" }}{PARA 0 "" 0 "" {TEXT -1 60 "Again, we can evaluate some integrals containing the Wright " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 92 " function , by a change of variables. Maple hasn't yet been taught this change \+ of variables." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "restart;\na lias(omega=RootOf(y+ln(y)=z,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "F := Int( sin(omega), z );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "student[changevar]( z=w+ln(w), F, w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(RootOf(_Z+ln(_Z)-w-ln(w))=w,%) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "In fact, this is perfectly general:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "F := Int( f(omega), z );" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "student[changevar](z=ln(w)+ w,F,w):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(RootOf(_Z+l n(_Z)-w-ln(w))=w,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "exp and(%);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 32 "Laplace Transform of t he Wright " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 9 " functio n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(inttrans):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "laplace( omega, z, s );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F := Int( omega*exp(-s*z), z );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "student[changevar](z =ln(w)+w, F, w):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "subs(Ro otOf(_Z+ln(_Z)-w-ln(w))=w,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Laplace_integral_omega := simplify(value(expand(simplify(%)))); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "I don't know how to simplify that any further! When z=0, we have omega=W(1); when z=infinity, we \+ have omega=infinity. But, Maple has difficulty evaluating the limits. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "?WhittakerM" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "L[omega] := limit( Laplace_integral _omega, w=infinity) - limit( Laplace_integral_omega, w=W(1));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 11 "Computation" }}{PARA 0 "" 0 "" {TEXT -1 97 "As before, \+ we have a family of arbitrary-order numerical iterative methods to com pute the Wright " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 10 " \+ function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "restart;\nalias( omega=RootOf(y+ln(y)-z,y),\n w = LambertW(exp(a)) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "s eries( omega, z=a, 4 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "fourth := (w,z) -> (w + w/(1+w)*(z-w-ln(w)) + w/(1+w)^3*(z-w-ln(w))^2 /2 - w*(2*w-1)/(1+w)^5*(z-w-ln(w))^3/6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "z := rand()/Float(1,12);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "computed_omega[0] := z;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "Digits := 1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "for i to 3 do\n Digits := 4*Digits; \n computed_omega[i] := \+ fourth( computed_omega[i-1], z );\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Digits := Digits + 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "z - computed_omega[3] - ln(computed_omega[3]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 18 "Concluding Remarks" }}{PARA 0 "" 0 "" {TEXT -1 30 "Than k you for listening to me " }{TEXT 265 6 "obsess" }{TEXT -1 175 "; I h ope that you have enjoyed some of these results, and I also hope that \+ you will feel, when you see W in a solution to one of your problems, t hat you have an actual answer." }}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 2 1 1805 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }