golden function

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

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

In functional form,

\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 -gold(-x). Both gold(x) and -gold(-x) furnish solutions to the equation

a-x-1/a=0

or, upon multiplying through by a,

a^2-xa-1=0.

Applying the quadratic equation to the above quadratic in a makes it immediately obvious that gold(x) furnishes the positive root of the equation, with -gold(-x) giving the negative solution. gold(1) gives 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)

See also: * hyperbolic functions

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