OFFSET
0,1
COMMENTS
At this value both sides of the equation x^e = e^(-x) equal: 0.46909728... .
Within the more general family of equations x^c = c^(-x), the solution for x is smallest when c = e.
Let's name cmin this constant: 0.7569451...
The general equation x^c = c^(-x) has real solutions only where c >= cmin.
At values of c in the range cmin < c < 1, there are two solutions.
At values of c < cmin, the two curves do not intersect.
At values of c =~ cmin, the two curves become essentially parallel over an extended range.
When c = cmin, x = e is the tangent point, where both sides of the equation equal exp(cmin) = 2.1317539... = cmin^(-e) = 1/0.46909728...
If the equation were x^e = e^x, the solution would be x = e.
FORMULA
From Gleb Koloskov, Aug 25 2021: (Start)
Equals Sum_{n>0} (-n/e)^(n-1)/n!. (End)
EXAMPLE
0.7569451064575836645840170881202415000611....
MATHEMATICA
RecurrenceTable [{a[n + 1] == N[1/Exp[a[n]]^(1/Exp[1]), 150],
a[1] == 3/4}, a, {n, 1, 200}]
RealDigits[ E*ProductLog[1/E], 10, 111][[1]] (* Robert G. Wilson v, Jan 30 2015 *)
PROG
(PARI) solve(x=0, 1, x^exp(1) - exp(-x)) \\ Michel Marcus, Dec 01 2014
(PARI) exp(1)*lambertw(exp(-1)) \\ Gleb Koloskov, Aug 25 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Richard R. Forberg, Dec 01 2014
STATUS
approved