Monogenic functions in finite-dimensional commutative associative algebras

Анотація

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over
the field of complex numbers with $m$ idempotents. Let $e_1=1$,\break $e_2,\ldots,e_k$, $2\leq
k\leq 2n$, are linearly independent over the field of real numbers elements of $\mathbb{A}_n^m$.
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 obtain a
constructive description of all mentioned functions by means of holomorphic functions of complex
variables. Due to this description obtain, that monogenic functions have Gateaux derivatives of
all orders. The present article is a generalization of the author's paper \cite{Shpakivskyi-2014},
where mentioned results are obtained for $k=3$.