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,

a² − 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

$\backslash operatorname\{arcsinh\}\backslash \; x=\; \backslash ln\; \backslash left\; (\; \backslash operatorname\{gold\}\backslash \; 2x\; \backslash right)$

and also satisfies the identity

gold  (x) ⋅ gold  (−x) = 1.

See also

• Hyperbolic functions