Asymptotic Behavior of the Hyperbolic Schwarz Map at Irregular Singular Points

概要

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Geometric study of a second-order Fuchsian differential equation u″−q(x)u = 0, where q is rational in x, has been made via the Schwarz map as well as via the hyperbolic and the derived Schwarz maps ([SYY]). When the equation admits an irregular singularity, such a study was first made in [SY] treating the confluent hypergeometric equation and the Airy equation. In this paper, we study the hyperbolic Schwarz map (note that this map governs the other Schwarz maps) of such an equation with any irregular singularity. We describe the asymptotic behavior of the map around the singular point: when the Poincaré rank is generic, it admits a uniform description; when the Poincaré rank is exceptional, a detailed study is made.