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We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat [2]. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann-Roch theorem. The proofs use energy estimates inspired by electrical networks [3].
This is a joint work with A. Bobenko [1].
Submitted - 2012 - http://arxiv.org/abs/1210.0561.
[2] C. Mercat, Discrete Riemann surfaces and the Ising model Comm. Math. Phys. - 2001. - 218:1. - 177-216.
[3] M. Skopenkov, Boundary value problem for discrete analytic functionsSubmitted - 2011 - http://arxiv.org/abs/1110.6737.