Curvilinear integral theorems for monogenic functions in commutative associative algebras

TL;DR

This paper extends classical complex analysis theorems to monogenic functions in finite-dimensional commutative associative algebras, establishing curvilinear integral theorems analogous to Cauchy's theorems.

Contribution

It introduces curvilinear integral theorems for monogenic functions in such algebras, generalizing fundamental complex analysis results.

Findings

Proves curvilinear Cauchy integral theorem for monogenic functions

Establishes Morera theorem in the algebraic context

Derives Cauchy integral formula for monogenic functions

Abstract

We consider an arbitrary finite-dimensional commutative associative algebra, $\mathbb{A}_n^m$, with unit over the field of complex number with $m$ idempotents. Let $e_1=1,e_2,e_3$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.e. continuous and differentiable in the sense of Gateaux) functions of the variable $xe_1+ye_2+ze_3$, where $x,y,z$ are real. For mentioned monogenic function we prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula.