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 .