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Actes des rencontres du CIRM

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Alexander I. Bobenko; Felix Günther
Discrete complex analysis – the medial graph approach
Actes des rencontres du CIRM, 3 no. 1: Discrete curvature: Theory and applications (2013), p. 159-169, doi: 10.5802/acirm.65
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Class. Math.: 39A12, 30G25
Keywords: Discrete complex analysis, quad-graphs, medial graph, Green’s identities, Cauchy’s integral formulae

Résumé - Abstract

We discuss a new formulation of the linear theory of discrete complex analysis on planar quad-graphs based on their medial graphs. It generalizes the theory on rhombic quad-graphs developed by Duffin, Mercat, Kenyon, Chelkak and Smirnov and follows the approach on general quad-graphs proposed by Mercat. We provide discrete counterparts of the most fundamental objects in complex analysis such as holomorphic functions, differential forms, derivatives, and the Laplacian. Also, we discuss discrete versions of important fundamental theorems such as Green’s identities and Cauchy’s integral formulae. For the first time, Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function are discretized.

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