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Main Title: Discrete complex analysis on planar Quad-graphs
Author(s): Bobenko, Alexander I.
Günther, Felix
Type: Book Part
Language Code: en
Is Part Of: 10.1007/978-3-662-50447-5
Abstract: We develop further a linear theory of discrete complex analysis on general quad-graphs, extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on the medial graph leads to generalizations as well as to new proofs of previously known discrete analogs of classical theorems. New results include in particular discretizations of Green’s first identity and Cauchy’s integral formula for the derivative of a holomorphic function. Another contribution is a discussion on the product of discrete holomorphic functions that is itself discrete holomorphic in a specific sense. In this paper, we focus on planar quad-graphs, but many notions and theorems can be easily adapted to discrete Riemann surfaces. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths explicit formulae for a discrete Green’s function and discrete Cauchy’s kernels are obtained. This slightly generalizes the previous results on rhombic lattices. When we further restrict to the integer lattice of a two-dimensional skew coordinate system a discrete Cauchy’s integral formulae for higher order derivatives is derived.
Issue Date: 2016
Date Available: 29-Aug-2017
DDC Class: 510 Mathematik
Subject(s): discrete complex analysis
green’s function
cauchy’s integral formulae
Book Title: Advances in discrete differential geometry
Editor: Bobenko, Alexander I.
Publisher: Springer
Publisher Place: Berlin, Heidelberg
Publisher DOI: 10.1007/978-3-662-50447-5_2
Page Start: 57
Page End: 132
ISBN: 978-3-662-50447-5
Appears in Collections:FG Geometrie und Integrable Systeme » Publications

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