The Cauchy-Pompeiu integral formula in elliptic complex numbers

TL;DR

This paper generalizes the Cauchy-Pompeiu integral formula to functions valued in elliptic algebras with a parameter-dependent structure polynomial, leading to a new integral representation for generalized holomorphic functions.

Contribution

It introduces a generalized Cauchy-Pompeiu formula for elliptic algebras with structure polynomial $X^2 + eta X + heta$, expanding the scope of complex analysis techniques.

Findings

Derived a generalized integral formula for elliptic algebra-valued functions.

Established a Cauchy integral representation for a broader class of holomorphic functions.

Extended classical complex analysis results to parameter-dependent elliptic algebras.

Abstract

The aim of this article is to give a generalization of the Cauchy-Pompeiu integral formula for functions valued in parameter-depending elliptic algebras with structure polynomial $X^2 + \beta X + \alpha$ where $\alpha$ and $\beta$ are real numbers. As a consequence, a Cauchy integral representation formula is obtained for a generalized class of holomorphic functions.