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Subject:Re: Is there an exact answer? e^x=x^2Author:Keith Geddes <kogeddes@scg.math.uwaterloo.ca>Organization:University of WaterlooDate:16 Feb 2001 17:26:56 GMT In article <J8r$QAAzFPj6Ewbl@quarksoft.demon.co.uk>, David Wilkinson <David@quarksoft.demon.co.uk> wrote: >In article <RW1j6.347$r1.167120@news.pacbell.net>, paradox ><mascota1@pacbell.net> writes >>a friend of mine showed me this question, and >>it didn't look all that complicated to solve, but >>it proved to be quite difficult. I have tried time >>and time again to solve this think. >> >>e^2=x^2 >> >>I know there is answer which I can get numerically, >>but is there an exact answer, and if so how would >>I go about getting it. >> >>Thanks! >> >> >For e^x=x^2 a quick look at the graphs of e^x and x^2 against x in >MathCad shows there is only one real root at about x = -0.70347 which, >as you say, you can get to any accuracy numerically. As with most >problems there probably is no simple exact answer. Answers in terms of >complicated functions are little help as you still have to evaluate them >numerically. >-- >David Wilkinson Yes -- except that what is a "known function" that people like to see (such as sin, exp, ln, BesselJ, etc.) can change over time with one's level of exposure to mathematics. The above question has a closed-form answer in terms of the LambertW function, as Maple will express. So to the question "Is there an exact answer?" we can answer: Yes. It leads one to realize that such a question "Is there an exact answer?" is entirely dependent on which set of functions you are willing to accept into your repertoire of "known functions". Is the LambertW function "worse" than exp, ln, etc.? Well not really -- except that its exposure in the mathematical literature has been relatively recent. As with the logarithm function, it has a very interesting behaviour over the complex field, with branch cuts and all that great stuff. The help page for LambertW in Maple tells me the following reference. - Reference: R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth. "On The Lambert W Function". Advances in Computational Mathematics 5 (1996): 329-359. And now one can find this function popping up in computational physics publications, etc. For the above equation in Maple -- ------------------------------------- |\^/| Maple 6 (SUN SPARC SOLARIS) ._|\| |/|_. Copyright (c) 2000 by Waterloo Maple Inc. \ MAPLE / All rights reserved. Maple is a registered trademark of <____ ____> Waterloo Maple Inc. | Type ? for help. > eqn := exp(x) = x^2; 2 eqn := exp(x) = x > soln := solve(eqn, x); soln := -2 LambertW(-1/2), -2 LambertW(1/2) > evalf(soln); 1.588047265 - 1.540223501 I, -.7034674224 # # So one of the given roots is complex and one is real. # # NOTE: The defining equation for the LambertW function is that it satisfies # # W(x) * exp(W(x)) = x # # > solve(y*exp(y) = x, y); LambertW(x) # # P.S. The fact that the LambertW function was given a name implies that # there is no way to express solutions to this type of equation using more # well-known elementary functions. ----------------------------------------------- Professor Keith Geddes Symbolic Computation Group Department of Computer Science University of Waterloo Waterloo ON N2L 3G1 CANADA E-mail: kogeddes@uwaterloo.ca -----------------------------------------------

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