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Sigmoid Function

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The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function

y=1/(1+e^(-x)).
(1)

It has derivative

(dy)/(dx)=[1-y(x)]y(x)
(2)
=(e^(-x))/((1+e^(-x))^2)
(3)
=(e^x)/((1+e^x)^2)
(4)

and indefinite integral

intydx=x+ln(1+e^(-x))
(5)
=ln(1+e^x).
(6)

It has Maclaurin series

y(x)=sum_(n=0)^(infty)((-1)^nE_n(0))/(2n!)x^n
(7)
=sum_(n=0)^(infty)((-1)^(n+1)(2^(n+1)-1)B_(n+1))/((n+1)!)x^n
(8)
=1/2+1/4x-1/(48)x^3+1/(480)x^5-(17)/(80640)x^7+(31)/(1451520)x^9-...,
(9)

where E_n(x) is an Euler polynomial and B_n is a Bernoulli number.

It has an inflection point at x=0, where

y^('')(x)=-(e^x(e^x-1))/((e^x+1)^3)=0.
(10)

It is also the solution to the ordinary differential equation

(dy)/(dx)=y(1-y)
(11)

with initial condition y(0)=1/2.

The logistic sigmoid is implemented in the Wolfram Language as LogisticSigmoid.

See also

Einstein Functions, Exponential Function, Exponential Ramp, Heaviside Step Function, Logistic Distribution, Logistic Equation

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References

Grebenc, A. "Foundations for a General Theory of Sigmoid Functions: Modelling with 30 Methods." In Intelligent Systems and Applications: Proceedings of the 2025 Intelligent Systems Conference (IntelliSys), Volume 1 (Ed. K. Arai). Cham, Switzerland: Springer, pp. 481-511, 2025. https://doi.org/10.1007/978-3-031-99958-1_31.von Seggern, D. CRC Standard Curves and Surfaces with Mathematics, 2nd ed. Boca Raton, FL: CRC Press, 2007.

Referenced on Wolfram|Alpha

Sigmoid Function

Cite this as:

Weisstein, Eric W. "Sigmoid Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SigmoidFunction.html

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