The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function
(1)
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It has derivative
(2)
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(3)
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(4)
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(5)
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(6)
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It has Maclaurin series
(7)
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(8)
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(9)
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where is an Euler polynomial and is a Bernoulli number.
It has an inflection point at , where
(10)
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It is also the solution to the ordinary differential equation
(11)
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with initial condition .