Is there a function which would be a "functional square root" of the exponential function? I.e., function $f(x)$ such that: \begin{equation} f(f(x))\equiv f^2(x)=e^x \end{equation}
If not, what is the proof of non-existence?
If there is such a function, is it unique? Is it analytic? If it isn't unique and can be analytic, is analytic solution unique?
