Here is the function:
$$f(a)=\sqrt{f(a)+\sqrt{f(f(a))+\sqrt{f(f(f(a)))+\cdots}}}$$ Is there another way to represent this function so that it only has $f(a)$ on one side and no $f(a)$'s on the other side of the $=$ sign? In other words, simplify the function as best as you can. Thank you very much.
Square both sides, rearrange and simplify:
$f(a) \cdot (f(a) - 1) - f(f(a)) = 0$
I doubt there is any way to get rid of the $f(f(a))$ term. And it may introduce extraneous solutions because of the squaring.