{"paper":{"title":"Scaling limits of discrete holomorphic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Guangbin Ren, Zeping Zhu","submitted_at":"2016-06-01T05:38:02Z","abstract_excerpt":"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.\n  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,\n  \\begin{eqnarray*} f(\\zeta)=\\displaystyle{\\int_{\\partial B^h}} \\mathcal{K}^h(z,\\zeta) f(z)dS^h(z)+\\displaystyle{\\int_{B^h}} E^h(\\zeta-z) \\partial_{\\bar z}^h f (z)dV^h(z),\\end{eqnarray*} which results "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.00121","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}