# S函数

S函數的曲線圖形
S函數在複數域的分布圖形

${\displaystyle S(t)={\frac {1}{1+e^{-t}}}.}$

${\displaystyle s:=1/2+{\frac {1}{4}}t-{\frac {1}{48}}t^{3}+{\frac {1}{480}}t^{5}-{\frac {17}{80640}}t^{7}+{\frac {31}{1451520}}t^{9}-{\frac {691}{319334400}}t^{11}+O(t^{12})}$

## 常見的S函數

${\displaystyle f(x)={\frac {1}{1+e^{-x}}}}$
${\displaystyle f(x)=\tanh x={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}}$
${\displaystyle f(x)=\arctan x}$
${\displaystyle f(x)=\operatorname {gd} (x)=\int _{0}^{x}{\frac {1}{\cosh t}}\,dt=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)}$
${\displaystyle f(x)=\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt}$
${\displaystyle f(x)=(1+e^{-x})^{-\alpha },\quad \alpha >0}$
${\displaystyle f(x)={\begin{cases}\displaystyle {\frac {\int _{0}^{x}{\bigl (}1-u^{2}{\bigr )}^{N}\ du}{\int _{0}^{1}{{\bigl (}1-u^{2}{\bigr )}^{N}\ du}}},&|x|\leq 1\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\,\quad N\geq 1}$
${\displaystyle f(x)={\frac {x}{\sqrt {1+x^{2}}}}}$

## 参考资料

• Mitchell, Tom M. Machine Learning. WCB–McGraw–Hill. 1997. ISBN 0-07-042807-7.. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logistic function" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – and the sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural nets.
• Humphrys, Mark. Continuous output, the sigmoid function. [2015-02-01]. （原始内容存档于2015-02-02）. Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.