Sigmoid Function

Min   Max       Re       Im

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)

and indefinite integral

			(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 .

REFERENCES:

von Seggern, D. CRC Standard Curves and Surfaces with Mathematics, 2nd ed. Boca Raton, FL: CRC Press, 2007.