Golden function

In mathematics, the golden function is the upper branch of the hyperbola

:$frac\{y^2-1\}\; \{y\}=x.$

In functional form,

:$y=operatorname\{gold\}\; x=\; frac\{x+sqrt\{x^2+4\; \{2\}.$

Once gold("x") has been defined, the lower branch of the hyperbola can also be defined as "y" = −gold(−"x"). Both gold("x") and −gold(−"x") furnish solutions for "a" of the equation

:$a-x-1/a=0\; ,$

or, upon multiplying through by "a",

:$a^2-xa-1=0.\; ,$

Applying the quadratic formula to the above quadratic equation in "a" shows that gold("x") is the positive root of the equation and −gold(−"x") is the negative solution. The value of gold(1) is the golden ratio and gold(2) gives the silver ratio 1 + √2.

The golden function is connected to the hyperbolic sine by the identity

: $operatorname\{arcsinh\}\; x=\; ln\; left\; (\; operatorname\{gold\}\; 2x\; ight)$

and also satisfies the identity

: $operatorname\{gold\}\; (x)\; cdot\; operatorname\{gold\}\; (-x)\; =\; 1.$

ee also

* Hyperbolic functions