Discrete complex analysis – the medial graph approach

Alexander I. Bobenko; Felix Günther

doi:10.5802/acirm.65

¹ Institut für Mathematik MA 8-4 Technische Universität Berlin Straße des 17. Juni 136 10623 BERLIN GERMANY
² Institut für Mathematik MA 8-3 Technische Universität Berlin Straße des 17. Juni 136 10623 BERLIN GERMANY

Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 159-169.

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.

Published online: 2014-11-12

Zbl: 06938613

DOI: 10.5802/acirm.65

Classification: 39A12, 30G25
Keywords: Discrete complex analysis, quad-graphs, medial graph, Green’s identities, Cauchy’s integral formulae

Author's affiliations:

Alexander I. Bobenko ¹; Felix Günther ²

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Alexander I. Bobenko; Felix Günther. Discrete complex analysis – the medial graph approach. Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 159-169. doi : 10.5802/acirm.65. https://acirm.centre-mersenne.org/articles/10.5802/acirm.65/

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