p+ , p– , x(T) , f(x)

Form the $x^2-xT-1=0$ where $T\in\mathbb{R}^{+}\ ;\ T≠0$
$x-\frac1x=T\ →\ \frac xT=\frac{xT+1}{xT}=\frac{x+\frac1T}x$
Def.: $x≡p^+$ and $\frac1x≡p^-$
$p^+=\frac{\sqrt{T^2+4}+T}2\ →\ p^-=\frac1{p^+}=\frac2{\sqrt{T^2+4}+T}=\frac{\sqrt{T^2+4}-T}2$

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Def.: $x\left({T}\right)=T+p^-\left({\frac1x}\right)=T+\frac{\sqrt{T^2+4}-T}2=\frac{\sqrt{T^2+4}+T}2$ Def.: $f\left({x}\right)≡f\left({x_T}\right)≡x\left({T}\right)$

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The properties of the $f\left({x}\right)$ :

1. $f\left({-x}\right)=\frac1{f\left({x}\right)}=f\left({x}\right)-x$
   $\frac1{f\left({-x}\right)}=f\left({x}\right)=f\left({-x}\right)+x$
2. $f\left({-x}\right)+x=\frac1{f\left({x}\right)-x}$
3. $x·f\left({x}\right)-x·f\left({-x}\right)=x^2$
   $x·f\left({-x}\right)-x·f\left({x}\right)=-x^2$