Integral theorems for monogenic functions in commutative algebras

Abstract

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ 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 $\sum_{j=1}^k x_j\,e_j$  where $x_1,x_2,\ldots,x_k$ are real, and prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula in $k$-dimensional ($2\leq k\leq
2n$)  real subspace of the algebra $\mathbb{A}_n^m$. The present  results are generalizations of the  corresponding results obtained in [1] for the case $k=3$.