Editing Logistic function (section)

===Hyperbolic tangent===

The logistic function is an offset and scaled [[hyperbolic tangent]] function:
<math display="block">f(x) = \frac12 + \frac12 \tanh\left(\frac{x}{2}\right),</math>
or
<math display="block">\tanh(x) = 2 f(2x) - 1.</math>

This follows from
<math display="block">
\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)} \\
         &= f(2x) - \frac{e^{-2x}}{1 + e^{-2x}} = f(2x) - \frac{e^{-2x} + 1 - 1}{1 + e^{-2x}} = 2f(2x) - 1.
\end{align}
</math>

The hyperbolic-tangent relationship leads to another form for the logistic function's derivative:

<math display="block">\frac{d}{dx} f(x) = \frac14 \operatorname{sech}^2\left(\frac{x}{2}\right),</math>

which ties the logistic function into the [[logistic distribution]].

Geometrically, the hyperbolic tangent function is the [[hyperbolic angle]] on the [[unit hyperbola]] <math>x^2 - y^2 = 1</math>, which factors as <math>(x + y)(x - y) = 1</math>, and thus has asymptotes the lines through the origin with slope {{tmath|-1}} and with slope {{tmath|1}}, and vertex at {{tmath|(1, 0)}} corresponding to the range and midpoint ({{tmath|0/1 = 0}}) of tanh. Analogously, the logistic function can be viewed as the hyperbolic angle on the hyperbola <math>xy - y^2 = 1</math>, which factors as <math>y(x - y) = 1</math>, and thus has asymptotes the lines through the origin with slope {{tmath|0}} and with slope {{tmath|1}}, and vertex at {{tmath|(2, 1)}}, corresponding to the range and midpoint ({{tmath|1/2}}) of the logistic function.

Parametrically, [[hyperbolic cosine]] and [[hyperbolic sine]] give coordinates on the unit hyperbola:{{efn|Using {{tmath|t}} for the parameter and {{tmath|(x, y)}} for the coordinates.}} <math>\left( (e^t + e^{-t})/2, (e^t - e^{-t})/2\right)</math>, with quotient the hyperbolic tangent. Similarly, <math>\bigl(e^{t/2} + e^{-t/2}, e^{t/2}\bigr)</math> parametrizes the hyperbola <math>xy - y^2 = 1</math>, with quotient the logistic function. These correspond to [[linear transformations]] (and rescaling the parametrization) of [[Hyperbola#As_an_affine_image_of_the_hyperbola_y_=_1/x|the hyperbola <math>xy = 1</math>]], with parametrization <math>(e^{-t}, e^t)</math>: the parametrization of the hyperbola for the logistic function corresponds to <math>t/2</math> and the linear transformation <math>\bigl( \begin{smallmatrix}
1 & 1\\ 0 & 1 \end{smallmatrix} \bigr)</math>, while the parametrization of the unit hyperbola (for the hyperbolic tangent) corresponds to the linear transformation <math>\tfrac{1}{2}\bigl( \begin{smallmatrix}
1 & 1\\ -1 & 1 \end{smallmatrix} \bigr)</math>.