Potential theory on a rhombic lattice
Potential theory on a rhombic lattice
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菱形晶格的势理论
DOI：
10.1016/s0021-9800(68)80072-9
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发表时间：
1968
期刊：
影响因子：
--
通讯作者：
R. Duffin
中科院分区：
文献类型：
--
作者：
R. Duffin
所关心的是复平面的格点上定义的复值函数。晶格可以是正方形块的通常晶格，但主要感兴趣的是用菱形代替正方形的不规则晶格。一个函数被定义为离散解析的，如果菱形的一条对角线上的差商等于另一条对角线上的差商。在此基础上，发展了经典函数论中下列概念的离散类似物：拉普拉斯方程、柯西-黎曼方程、微分、围道积分、莫雷拉定理和调和多项式。这个理论不仅仅是一个类比，因为对于一类常见的边值问题，它证明了可以获得离散调和函数的经典Dirichlet积分的上下界。
Of concern are complex valued functions defined on the lattice points of the complex plane. The lattice can be the usual lattice of square blocks but of main interest is an irregular lattice with squares replaced by rhombs. A function is defined to be discrete analytic if the difference quotient across one diagonal of a rhomb equals the difference quotient across the other diagonal. Based on this definition discrete analogs of the following concepts in classical function theory are developed: Laplace equation, Cauchy-Riemann equations, differentiation, contour integration, Morera's theorem, and harmonic polynomials. The theory is more than an analogy because, for a common class of boundary value problems, it proves possible to obtain upper and lower bounds for the classical Dirichlet integral in terms of discrete harmonic functions.