A364931 Decimal expansion of the solution to 1 / (1 + e^(-x)) = x.

6, 5, 9, 0, 4, 6, 0, 6, 8, 4, 0, 7, 4, 0, 6, 6, 6, 0, 9, 8, 4, 3, 8, 6, 4, 9, 5, 9, 2, 8, 8, 6, 7, 5, 5, 1, 6, 9, 8, 0, 9, 0, 3, 3, 0, 3, 5, 7, 1, 1, 5, 1, 8, 8, 4, 8, 9, 2, 9, 4, 6, 3, 0, 8, 0, 4, 2, 2, 4, 1, 1, 3, 0, 4, 2, 4, 2, 9, 7, 5, 7, 5, 7, 9, 0, 3, 4, 7, 3, 5, 6, 4, 8, 2, 8, 2, 0, 1, 0, 5

Offset: 0

Author

Michal Paulovic, Aug 13 2023

Keywords

• nonn
• cons

Comments

Fixed point of sigmoid (standard logistic function) as well as its inverse (logit).

Examples

			0.6590460684074066...
		

Links

• Eric Weisstein's World of Mathematics, Sigmoid Function.
• Wikipedia, Logistic Function.

Programs

Maple

Digits:=100: fsolve(1/(1+exp(-x))-x, x);

Mathematica

RealDigits[FindRoot[1/(1+Exp[-x])-x, {x, 0.6}, WorkingPrecision -> 100][[1, 2]], 10, 100][[1]]

PARI

default(realprecision, 100); solve(x=0.4,0.7,1/(1+exp(-x))-x)