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 .