Re: Is there an exact answer? e^x=x^2 by Keith Geddes

Subject:      Re: Is there an exact answer? e^x=x^2
Author:       Keith Geddes <kogeddes@scg.math.uwaterloo.ca>
Organization: University of Waterloo
Date:         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
-----------------------------------------------