A generalization of the root function

TL;DR

This paper generalizes the concept of the n-th root function using inverse branches of n-factor Blaschke-products, providing explicit formulas, interpretations, and an algorithm for numerical construction on the disk and torus.

Contribution

It introduces a novel generalization of the root function via inverse branches of Blaschke-products, with explicit formulas and a new algorithm for numerical computation.

Findings

Explicit formulas for inverse branches on the torus

A finite-step algorithm for solution trajectories

Two natural interpretations of inverse branches

Abstract

We consider the interpretation and the numerical construction of the inverse branches of $n$ factor Blaschke-products on the disk and show that these provide a generalization of the $n$-th root function. The inverse branches can be defined on pairwise disjoint regions, whose union provides the disk. An explicit formula can be given for the $n$ factor Blaschke-products on the torus, which can be used to provide the inverses on the torus. The inverse branches can be thought of as the solutions $z=z_t(r) (0\le r\le 1)$ to the equation $B(z )=re^{it}$, where $B$ denotes an $n$ factor Blaschke-product. We show that starting from a known value $z_t(1)$, any $z_t(r)$ point of the solution trajectory can be reached in finite steps. The appropriate grouping of the trajectories leads to two natural interpretations of the inverse branches (see Figure 2). We introduce an algorithm which can be used to find the points of the trajectories.