Multidomain spectral method for the Gauss hypergeometric function

TL;DR

This paper introduces a multidomain spectral method combining Frobenius' approach and Moebius transformations to solve the hypergeometric differential equation with high accuracy across complex domains.

Contribution

It develops a novel hybrid spectral method that decomposes the domain and matches solutions at boundaries, enabling efficient and accurate solutions on the entire Riemann sphere.

Findings

Achieves high-precision solutions for hypergeometric functions.

Provides a framework for extending solutions to complex domains.

Demonstrates the method's effectiveness on the entire Riemann sphere.

Abstract

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius' method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line $\mathbb{R}\cup {\infty}$, except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier--ultraspherical spectral method.