Hyperbolic Schwarz Maps of the Airy and the Confluent Hypergeometric Differential Equations and Their Asymptotic Behaviors

The Schwarz map of the hypergeometric differential equation is studied first by Schwarz, and later by several authors for various generalizations of the hypergeometric equation. But up to now nothing is studied about the Schwarz map for confluent equations, mainly because such a map would produce just a chaos. Recently we defined the hyperbolic Schwarz map, and studied in several cases, including confluent hypergeometric ones, geometric properties of the image surfaces in the hyperbolic 3-space. In this paper, we first study the hypergeometric Schwarz map of the Airy equation, which can be regarded as the doubly confluent hypergeometric equation. The image surface has triangular cuspidal edge curve, and at the three vertices it has three swallowtails. We present some global behaviors by examining the asymptotic behavior of Airy functions at infinity. We next describe the asymptotic behavior of the hyperbolic Schwarz map of the confluent hypergeometric differential equation, which includes the Bessel differential equation; we thus complement the previous study for the confluent hypergeometric equation in [SSY].