I am trying to solve the following question:
Show that there exists an analytic function $f$ in the open right half-plane such that $(f(z))^2 + 2f(z) \equiv z^2$. Show that your function $f$ can be continued analytically to a region containing the set $\{z \in \mathbb{C} , \lvert z \rvert = 3\}$.
I can solve the first part. Since $z^2+1$ is a non-vanishing holomorphic function in the open right half-plane, which is simply connected, the function admits an analytic square root $h(z)$ in this region; it then suffices to choose $h(z)-1$ as our required function $f(z)$.
I don't know how to proceed with the analytic continuation part though. Any suggestions?