Numerical evaluation of the Gauss hypergeometric function: Implementation and application to Schramm-Loewner evolution

TL;DR

This paper addresses the challenge of numerically evaluating the Gauss hypergeometric function in the context of Schramm-Loewner evolution, providing practical implementation methods for accurate computations in relevant parameter regimes.

Contribution

It introduces and compares approaches for numerical evaluation of ${}_2F_1$, offering a ready-to-use implementation suitable for SLE studies.

Findings

Developed a method for accurate ${}_2F_1$ evaluation in critical parameter ranges.

Identified limitations of existing software tools for this purpose.

Provided a practical implementation for researchers in the field.

Abstract

Numerical studies of fractal curves in the plane often focus on subtle geometrical properties such as their left passage probability. Schramm-Loewner evolution (SLE) is a mathematical framework which makes explicit predictions for such features of curve ensembles. The SLE prediction for the left passage probability contains the Gauss hypergeometric function ${}_2F_1$. To perform computational SLE studies it is therefore necessary to have a method for numerical evaluation of ${}_2F_1$ in the relevant parameter regime. In some instances, commercial software provides suitable tools, but freely available implementations are rare and are usually unable to handle the parameter ranges needed for the left passage probability. We discuss different approaches to overcome this problem and also provide a ready-to-use implementation of one conceptually transparent method.