Re: exp(x) + k x = 0; analytic solution

Jean-Marc Gulliet <jeanmarc.gulliet@xxxxxxxxx> writes:

Per Abrahamsen wrote:

I can easily convince myself that there is exactly one solution when k
0, but does it have a nice analytical solution, or do I have to use
numeric methods?

I apologize if this is the wrong place to ask.

The analytic solution returned by Mathematica 6.0.1 for k positive and x
real is in terms of *ProductLog* (principal value of the Lambert
W-function in Mathematica. See
http://mathworld.wolfram.com/LambertW-Function.html)

In[1]:= Reduce[{Exp[x] + k x == 0, k > 0}, x, Reals]

Out[1]= k > 0 && x == -ProductLog[1/k]

Or in Maple, x = -LambertW(1/k).
The LambertW function has infinitely many branches, but only the principal
branch gives a real solution when k > 0. When k < -e, there are two
real solutions given by the principal branch and the "-1" branch.
When k = -e, there is one real solution x=1 (common to both those branches).
When -e < k <= 0 there are no real solutions.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.