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A248200
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Decimal expansion of x in the solution to x^e = e^(-x), where e = exp(1). Also the smallest value of the constant c where there exists a solution to x^c = c^(-x).
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0
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7, 5, 6, 9, 4, 5, 1, 0, 6, 4, 5, 7, 5, 8, 3, 6, 6, 4, 5, 8, 4, 0, 1, 7, 0, 8, 8, 1, 2, 0, 2, 4, 1, 5, 0, 0, 0, 6, 1, 1, 2, 7, 6, 6, 0, 1, 8, 7, 3, 6, 5, 8, 0, 8, 2, 1, 0, 5, 2, 8, 7, 2, 7, 5, 4, 6, 5, 7, 1, 9, 7, 2, 4, 2, 8, 2, 6, 1, 9, 7, 9, 0, 2, 5, 0, 6, 5, 3, 5, 8, 5, 6, 0, 6, 5, 2, 2, 0, 7, 7, 6, 4, 7, 1, 6, 8, 1, 2, 0
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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Equals Sum_{n>0} (-n/e)^(n-1)/n!. (End)
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EXAMPLE
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0.7569451064575836645840170881202415000611....
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MATHEMATICA
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RecurrenceTable [{a[n + 1] == N[1/Exp[a[n]]^(1/Exp[1]), 150],
a[1] == 3/4}, a, {n, 1, 200}]
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PROG
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(PARI) solve(x=0, 1, x^exp(1) - exp(-x)) \\ Michel Marcus, Dec 01 2014
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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