The sigmoid function, or the sigmoidal curve (von Seggern 2007, p. 148) as well as logistic function, is the function:

It has derivative

) | |||

and indefinite integral

(5) | |||

(6) |

It has Maclaurin series

(7) | |||

(8) | |||

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

It has an inflection point at , where

(10) |

The same as the solution to the ordinary differential equation

(11) |

with initial condition .