Discrete complex analysis on planar quad-graphs

TL;DR

This paper develops a comprehensive linear theory of discrete complex analysis on planar quad-graphs, extending classical concepts and theorems to discrete settings, including new discretizations of Green's identity and Cauchy's formula.

Contribution

It introduces a unified framework for discrete complex analysis on quad-graphs, with new discretizations of fundamental theorems and explicit constructions of Green's functions and Cauchy's kernels.

Findings

Discrete Green's function constructed for parallelogram-graphs

Discrete Cauchy's kernels with smooth asymptotics

Discrete Cauchy's integral formulae for higher derivatives

Abstract

We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and 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 yields more instructive proofs of discrete analogs of several classical theorems and even new results. We provide discrete counterparts of fundamental concepts in complex analysis such as holomorphic functions, derivatives, the Laplacian, and exterior calculus. Also, we discuss discrete versions of important basic theorems such as Green's identities and Cauchy's integral formulae. For the first time, we discretize Green's first identity and Cauchy's integral formula for the derivative of a holomorphic function. In this paper, we focus on planar quad-graphs, but we would like to mention that many notions and theorems can be adapted to discrete Riemann surfaces in a straightforward way. In the case of planar parallelogram-graphs with bounded interior angles and bounded ratio of side lengths, we construct a discrete Green's function and discrete Cauchy's kernels with asymptotics comparable to the smooth case. Further restricting to the integer lattice of a two-dimensional skew coordinate system yields appropriate discrete Cauchy's integral formulae for higher order derivatives.