Golden function
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In mathematics, the golden function is the upper branch of the hyperbola
- <math> \frac{y^2-1} {y}=x.<math>
In functional form,
- <math> y=\operatorname{gold}\ x= \frac{x+\sqrt{x^2+4}} {2}. <math>
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
- <math> a-x-1/a=0 \,<math>
or, upon multiplying through by a,
- <math> a^2-xa-1=0. \,<math>
Applying the quadratic formula to the above quadratic equation in a makes it immediately obvious that gold(x) is the positive root of the equation and −gold(−x) is 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
- <math> \operatorname{arcsinh}\ x= \ln \left ( \operatorname{gold}\ 2x \right).
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