Sigmoid Function

$\begin{figure}\begin{center}\BoxedEPSF{SigmoidFunction.epsf scaled 700}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{SigmoidFunctionReIm.epsf scaled 800}\end{center}\end{figure}$

The function

$\begin{displaymath} y={1\over 1+e^{-x}} \end{displaymath}$

which is the solution to the Ordinary Differential Equation

$\begin{displaymath} {dy\over dx}=y(1-y). \end{displaymath}$

It has an inflection point at

, where

$\begin{displaymath} y''(x)=-{e^x(e^x-1)\over(e^x+1)^3}=0. \end{displaymath}$

References

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 124, 1993.