ar X iv :m at h/ 04 02 09 7v 1 [m ath .D G] 6 Fe b 2 00 4 Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function Alexander I. Bobenko ? Christian Mercat † Yuri B. Suris ‡ February 1, 2008 Abstract. Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The linear theory is based on the discrete Cauchy-Riemann equations, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable (in the sense of 3D consistency, which yields also the existense of zero curvature representations, Backlund transformations etc.). We demonstrate that in some precise sense the linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions which take real (imaginary) values on two sublattices. We extend solutions of the basic equations of both theories to Zd, where d is the number of different edge slopes of the quasicrystallic tiling. In the linear theory, we give an integral representation of an arbitrary discrete holomorphic function, thus proving the density of discrete exponential functions.
- g?
- embedded quad-graph
- graph called
- cauchy-riemann equations
- g? dual
- discrete holomorphic
- cross-ratio equations
- quad-graph