A while back I got very interested in limits of the form
$$ \lim_{n\to\infty} (2A)^n \left (A-\underbrace{\sqrt{a+\sqrt{a+\ldots\sqrt{a+z}}}}_{n\textrm{ radicals}} \right )=f_a^{-1}(z) $$
Where $A$ is the positive solution of $A^2=a+A$. As notated, the limit can be used to define a function with some rather interesting properties. I explore some of them here. In particular, the function becomes trigonometric in the case $a=2$, and related to the golden ratio in the case $a=1$. These values also correspond to notable properties of the function. In particular, the roots of the function have some very interesting behavior that seems to become fractal.
Does anyone know if this limit has been studied before? It seems to be related to the Mandelbrot set, inasmuch as the iterated radical is the inverse operation of the Mandelbrot iteration, though I haven't looked into this connection very much.
Any information about other research on this subject or related fields of study would be greatly appreciated.
