hi,
I have some problem with the proof of the following problem
sigmoid function ($\displaystyle r(s)= 1/(1+e^-^s)$)
is antisymetric about the point (0,1/2)
help me please!!!
Anti-symmetric about $\displaystyle (0,1/2)$ means that $\displaystyle t(s)=r(s)-1/2$ is anti-symmetric, that is:Originally Posted by EITP
$\displaystyle
t(s)=-t(-s)
$
Now:
$\displaystyle
t(s)=\frac{1}{1+e^{-s}}-\frac{1}{2}=\frac{1-e^{-s}}{2(1+e^{-s})}
$.
Now multiply both top and bottom of the LHS by $\displaystyle e^{s}$:
$\displaystyle
t(s)=\frac{e^s-1}{2(e^s+1)}=-t(-s)
$
QED
RonL