## Abstract

We study analytic solutions of the functional equation $h(z)^2-h(z^2)+c = 0,$ where $c > 0$ is a parameter, and constant solutions are excluded. It suffices to consider solutions for which $h(z)-z^{-1}$ is regular in a neighbourhood of $z = 0$. If $0 < c \leqslant \frac{1}{4}, h(z)$ can be continued as a single-valued analytic function into the unit disc $|z| < 1$ where its only singularity is the pole at $z = 0$; the circle $|z| = 1$ is a natural boundary. On the other hand, if $c > \frac{1}{4}$, then by analytic continuation $h(z)$ becomes a multiple-valued function with an infinite sequence of quadratic branch points tending to every point of $|z| = 1$, and no branch of $h(z)$ can be continued beyond this circle. A change of variable transforms (H) into the difference equation $g(Z+1)-g(Z)=g(Z)^2+C,$ where $C$ is a real parameter. The solutions of this equation have properties similar to those of (H).