The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function
(1)

It has derivative
(2)
 
(3)
 
(4)

(5)
 
(6)

It has Maclaurin series
(7)
 
(8)
 
(9)

where is an Euler polynomial and is a Bernoulli number.
It has an inflection point at , where
(10)

It is also the solution to the ordinary differential equation
(11)

with initial condition .