A Geometric Study of the Hypergeometric Function with Imaginary Exponents

Abstract

The Schwarz map defined by the ratio of two solutions of the hypergeometric equation has been studied mainly when the exponents are real. In this paper, we study this map when the exponents are purely imaginary, a case that has been neglected for over a hundred years. A fundamental domain in the source plane and that in the target plane are constructed; the Schwarz map restricted on these domains is conformally isomorphic and the whole map can be recovered by this restriction through repeated use of the Schwarz reflection principle. We investigate the shape of these fundamental domains both analytically and numerically, and conclude with open questions.