# Logistic Sigmoid Function

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A Logistic Sigmoid Function is a sigmoid exponential probability function of the form $\displaystyle{ f(t,A,B,C) \equiv (C + Ae^{-Bt})^{-1} }$.

## References

### 2018

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/logistic_function#Mathematical_properties Retrieved:2018-2-4.
• The standard logistic function is the logistic function with parameters (k = 1, x0 = 0, L = 1) which yields : \displaystyle{ \begin{align} f(x) &= \frac{1}{1 + e^{-x}} \\ &= \frac{e^x}{1 + e^x} \\ &= \tfrac12 + \tfrac12 \tanh(\tfrac{x}{2}) \\ \end{align} } In practice, due to the nature of the exponential function ex, it is often sufficient to compute the standard logistic function for x over a small range of real numbers such as a range contained in [−6, +6].

The logistic function has the symmetry property that: : $\displaystyle{ 1-f(x) = f(-x). }$ Thus, $\displaystyle{ x \mapsto f(x) - 1/2 }$ is an odd function.

The logistic function is an offset and scaled hyperbolic tangent function : $\displaystyle{ f(x) = \tfrac12 + \tfrac12 \tanh(\tfrac{x}{2}) }$ or : $\displaystyle{ \tanh(x) = 2 \, f(2x) - 1 }$ .

This follows from : \displaystyle{ \begin{align} \tanh(x) & = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^x \cdot \left(1 - e^{-2x}\right)}{e^x \cdot \left(1 + e^{-2x}\right)} \\[6pt] & = f(2x) - \frac{e^{-2x}}{1+e^{-2x}} = f(2x) - \frac{e^{-2x} + 1 - 1}{1+e^{-2x}} = 2f(2x)-1. \end{align} }