Golden function

From Wikipedia, the free encyclopedia

The 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

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