An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type

• Ando Hisashi

  Kyushu University

• Hay Mike

  Roma Tre University

• Kajiwara Kenji

  Kyushu University

• Masuda Tetsu

  Aoyama Gakuin University

Abstract

We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric τ functions for the sixth Painlevé equation. The original definition of the discrete power function imposes strict conditions on the domain and the value of the exponent. However, we show that one can extend the value of the exponent to arbitrary complex numbers except even integers and the domain to a discrete analogue of the Riemann surface. Moreover, we show that the discrete power function is an immersion when the real part of the exponent is equal to one.

Journal

• Funkcialaj Ekvacioj

  Funkcialaj Ekvacioj 57 (1), 1-41, 2014

  Division of Functional Equations, The Mathematical Society of Japan

Keywords

• Discrete conformal mapping
• Hypergeometric function
• Painlevé VI equation
• Circle patterns
• Discrete differential geometry

Details

• CRID: 1390282680088448128
• NII Article ID: 130003391675
• DOI: 10.1619/fesi.57.1
• ISSN: 05328721
• Text Lang: en
• Data Source:
  • JaLC
  • Crossref
  • CiNii Articles
  • KAKEN
• Abstract License Flag: Disallowed