Experimental Mathematics

A Geometric Study of the Hypergeometric Function with Imaginary Exponents

Takeshi Sasaki and Masaaki Yoshida

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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.

Article information

Source
Experiment. Math., Volume 10, Issue 3 (2001), 321-330.

Dates
First available in Project Euclid: 25 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1069786340

Mathematical Reviews number (MathSciNet)
MR1917420

Zentralblatt MATH identifier
0998.33002

Keywords
Hypergeometric function hypergeometric equation Schwarzian derivative Schwarz $s$-function Schwarz reflection principle Schottky group connection matrix

Citation

Sasaki, Takeshi; Yoshida, Masaaki. A Geometric Study of the Hypergeometric Function with Imaginary Exponents. Experiment. Math. 10 (2001), no. 3, 321--330. https://projecteuclid.org/euclid.em/1069786340


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