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

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

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 exp...

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Veröffentlicht in:Funkcialaj Ekvacioj 2014, Vol.57(1), pp.1-41
Hauptverfasser: Ando, Hisashi, Hay, Mike, Kajiwara, Kenji, Masuda, Tetsu
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Sprache:eng
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Circle patterns
Discrete conformal mapping
Discrete differential geometry
Hypergeometric function
Painlevé VI equation
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container_title Funkcialaj Ekvacioj
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creator Ando, Hisashi
Hay, Mike
Kajiwara, Kenji
Masuda, Tetsu
description 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.
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subjects Circle patterns
Discrete conformal mapping
Discrete differential geometry
Hypergeometric function
Painlevé VI equation
title An Explicit Formula for the Discrete Power Function Associated with Circle Patterns of Schramm Type
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