# S函數

Sigmoid function 2D plot
Sigmoid function complex plot

S函數得名因其形狀像S字母。

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

S函數的級數展開為：

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

## 參考資料

• 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. Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.