An explicit formula for the discrete power function associated with circle patterns of Schramm type

Details 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

2011-05-09

arxiv.org/abs/1105.1612v3

Nonlinear Sciences

Exactly Solvable and Integrable Systems

33 pages. Change from the previous version: Section 5 has been added where the immersion property of the discrete power functi

Scientific paper

We present an explicit formula for the discrete power function introduced by Bobenko, which is expressed in terms of the hypergeometric \tau functions for the sixth Painlev\'e 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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