Scaling limits of discrete holomorphic functions

2016/06/01 by Guangbin Ren, Ren, Guangbin, Zeping Zhu +1
Mathematics · #30G25 39A12 49M25 31C20 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1606.00121

openalex publication_date 2016/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

One of the most natural and challenging issues in discrete complex analysis is to prove the convergence of discrete holomorphic functions to their continuous counterparts. This article is to solve the open problem in the general setting. To this end we introduce new concepts of discrete surface measure and discrete outer normal vector and establish the discrete Cauchy-Pompeiu integral formula, f(ζ)=∫_{∂ B^h} K^h(z,ζ) f(z)dS^h(z)+∫_B^h E^h(ζ-z) ∂ _z^h f (z)dV^h(z), which results in the uniform convergence of the scaling limits of discrete holomorphic functions up to second order derivatives in the standard square lattices.

Citations

• �ber die partiellen Differenzengleichungen der mathematischen Physik
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