Cauchy theorem for a surface integral in commutative algebras

TL;DR

This paper extends the classical Cauchy integral theorem to hyperholomorphic functions in three-dimensional domains within commutative Banach algebras, even with non-smooth boundaries.

Contribution

It introduces a generalized Cauchy theorem for hyperholomorphic functions in three-dimensional settings with non-piecewise smooth boundaries in commutative Banach algebras.

Findings

Proves the Cauchy integral theorem in a new algebraic context.

Handles domains with non-smooth boundaries.

Applies to functions valued in finite-dimensional commutative Banach algebras.

Abstract

We prove an analogue of the Cauchy integral theorem for hyperholomorphic functions given in three-dimensional domains with non piece-smooth boundaries and taking values in an arbitrary finite-dimensional commutative associative Banach algebra.