Fast computation and convergence analysis of the infinite-product representation of the Schottky--Klein prime function

Abstract

The Schottky--Klein prime function is a standard tool for boundary-value problems on multiply connected circular domains. Because this function is represented as an infinite product over a Schottky group, numerical evaluation requires truncation to finitely many factors. The standard word-length truncation grows exponentially in cost and becomes inefficient when the boundary circles nearly touch one another or the unit circle. To address this difficulty, we assign to each group element a cross-ratio potential measuring the size of its contribution, and retain only terms below a prescribed threshold. We establish uniform closed-form bounds on the change in this potential when prepending Schottky-group generators, and from these bounds we derive an efficient enumeration algorithm. The resulting relative error decays exponentially with the threshold at a rate determined by the Hausdorff dimension of the limit set of the Schottky group. Numerical experiments demonstrate that the proposed formulation achieves substantial computational speedups over word-length truncation in challenging geometric configurations.