A generalization of the root function

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.