The Ultra-Radical: Analytic Continuation, Branching, and Stability of the Principal Branch

We study the ultra-radical $\sqrt[{n;a;b}]{x}$, the multi-valued solution to $y^{a}=1+a x y^{b}$. Inside the convergence radius $|x|<R$, every branch is given by a Master-J power series; for $|x|\ge R$, analytic continuation requires switching to one of two conjugate series. We introduce a determini