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In this short paper, we present discrete Borel-Pompeiu and Cauchy formulae on a rectangular lattice for bounded domains in \(\mathbb {R}^{2}\) . At first, the discrete geometrical setting for interior problems is introduced. After that, the discrete Cauchy-Riemann operators and their fundamental solutions are defined. Finally, the discrete T-operator and discrete F-operator are presented. The results presented in this paper constitute the basis of discrete operator calculus on rectangular lattices, which can be used to solve boundary value problems, as well as studying boundary values of discrete holomorphic functions.

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Discrete Borel-Pompeiu and Cauchy Formulae on a Rectangular Lattice for Bounded Domains in \(\mathbb {R}^{2}\)

  • 会议论文

  • 发表日期：

    2025年

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  • Klaus Gürlebeck,
  • Anastasiia Legatiuk

摘要

In this short paper, we present discrete Borel-Pompeiu and Cauchy formulae on a rectangular lattice for bounded domains in \(\mathbb {R}^{2}\) . At first, the discrete geometrical setting for interior problems is introduced. After that, the discrete Cauchy-Riemann operators and their fundamental solutions are defined. Finally, the discrete T-operator and discrete F-operator are presented. The results presented in this paper constitute the basis of discrete operator calculus on rectangular lattices, which can be used to solve boundary value problems, as well as studying boundary values of discrete holomorphic functions.