Some extensive discussions of Liouville's theorem and Cauchy's integral theorem on structural holomorphic

2018/07/07 by Gen Wang, Wang, Gen
Mathematics · #30D20 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #math.CV #msc:30D20

paper · pdf · doi:10.48550/arxiv.1807.02600

14 pages

openalex publication_date 2018/07/07 · openalex created_date 2018/07/19 · arxiv created 2020/02/24 · arxiv updated 2020/02/25 · openalex updated_date 2026/07/28

Abstract

Classic complex analysis is built on structural function K=1 only associated with Cauchy-Riemann equations, subsequently various generalizations of Cauchy-Riemann equations start to break this situation. The goal of this article is to show that only structural function K=Const such that Liouville's theorem is held, otherwise, it's not valid any more on complex domain based on structural holomorphic, the correction should be w=Φe^-K, where Φ=Const. Those theories in complex analysis which keep constant are unable to be held as constant in the framework of structural holomorphic. Synchronously, it deals with the generalization of Cauchy's integral theorem by using the new perspective of structural holomorphic, it is also shown that some of theories in the complex analysis are special cases at K=Const, which are narrow to be applied such as maximum modulus principle.

Citations

• Generalizations of the Cauchy Integral Theorems
• Some Simplifications in Basic Complex Analysis
• Schwarz lemma and Kobayashi metrics for holomorphic and pluriharmonic functions
• Nonlinear Cauchy-Riemann Equations and Liouville Equation For Conformal Metrics
• Complex derivatives are continuous - three self-contained proofs. Part 1
• A proof of Liouville's theorem via o-minimality

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