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

Abstract

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 deterministic geometric criterion that selects, for each branch index $n$, the correct conjugate series, thereby eliminating heuristic search and guaranteeing branch continuity across $|x|=R$. Key finding: Only the principal branch ($n=0$) remains continuous when the parameters $a$, $b$, and $x$ vary smoothly. This includes the critical limits $a\to0$ (transition to an exponential equation) and $b\to0$ (transition to a binomial root), where the principal branch converges to the corresponding classical solution. In contrast, branches with $n\neq0$ exhibit oscillatory divergence as $a\to0$ and lose their identity in these limits. This structural continuity singles out the principal branch for applications where parameters may vary with the system&#39;s state, such as in nonlinear media with field-dependent exponents or adaptive dynamical systems.