Discrete complex analysis on planar quad-graphs
Discrete complex analysis on planar quad-graphs
复制标题
平面四边形图上的离散复分析
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Felix Gunther
中科院分区：
文献类型：
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作者：
A. Bobenko;Felix Gunther
我们进一步发展了一般四边形图上离散复形分析的线性理论，扩展了 Duffin、Mercat、Kenyon、Chelkak 和 Smirnov 先前在菱形四边形图上离散复形分析的工作。我们基于中值图的方法导致了概括以及先前已知的经典定理离散类似物的新证明。新结果特别包括格林第一恒等式和全纯函数导数的柯西积分公式的离散化。另一个贡献是对离散全纯函数的乘积的讨论，该函数本身在特定意义上是离散全纯的。在本文中，我们关注平面四边形图，但许多概念和定理可以很容易地适用于离散黎曼曲面。在具有有界内角和有界边长比的平面平行四边形图的情况下，获得了离散格林函数和离散柯西核的显式公式。这稍微概括了先前关于菱形晶格的结果。当我们进一步限制二维斜坐标系的整数格时，导出了高阶导数的离散柯西积分公式。
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.
影响因子：
1.7
作者：
Bobenko;Alexander I;Günther
通讯作者：
Günther