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

TL;DR

This paper investigates the conditions under which Liouville's and Cauchy's integral theorems hold in the framework of structural holomorphic functions, revealing that only constant structural functions preserve these classical results.

Contribution

It demonstrates that Liouville's theorem requires a constant structural function and generalizes Cauchy's integral theorem within the structural holomorphic framework, identifying limitations of classical theories.

Findings

Liouville's theorem holds only for constant structural functions

Cauchy's integral theorem is generalized using structural holomorphic functions

Classical theories like maximum modulus principle are special cases at constant structural functions

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 $w=\Phi {{e}^{-K}}$, where $\Phi =Const$. 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.