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

TL;DR

This paper investigates the properties of the ultra-radical multi-valued function, establishing a geometric criterion for branch selection, and finds that only the principal branch remains continuous under parameter variation, with implications for nonlinear systems.

Contribution

It introduces a deterministic geometric criterion for selecting the correct branch of the ultra-radical function, ensuring continuity and analyzing its stability across parameter changes.

Findings

Only the principal branch remains continuous under parameter variation.

Branches with n≠0 diverge or lose identity as parameters approach critical limits.

The principal branch converges to classical solutions in limiting cases.

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's state, such as in nonlinear media with field-dependent exponents or adaptive dynamical systems.