From hyperbolic to complex Euler integrals

TL;DR

This paper investigates the degeneration of hyperbolic hypergeometric integrals to complex hypergeometric functions, providing detailed analysis and bounds to understand their limiting behavior.

Contribution

It offers a detailed study of how hyperbolic hypergeometric integrals reduce to complex hypergeometric functions, including new bounds on integrands.

Findings

Hyperbolic beta integral degenerates to a complex plane integral.

Conical functions are shown to degenerate to two-dimensional complex integrals.

Uniform bounds on integrands facilitate the degeneration analysis.

Abstract

Hyperbolic hypergeometric integrals are defined as Barnes-type integrals of products of hyperbolic gamma functions. Their reduction to ordinary hypergeometric functions is well known. We study in detail their degeneration to complex hypergeometric functions. Namely, using uniform bounds on the integrands, we prove that the univariate hyperbolic beta integral and the conical function degenerate to two-dimensional integrals over the complex plane.