有限维交换结合代数中的单演函数
Monogenic functions in finite-dimensional commutative associative algebras
摘要：
设$\mathbb{A}_n^m$是复数域上具有$m$幂等元的任意$n$ -维交换结合代数.设具有$2\leq k\leq 2n$的$e_1=1,e_2,\ldots,e_k$是$\mathbb{A}_n^m$的在实数域上线性无关的元素。我们认为单基因（i.~ e.连续和可微的意义上的Gateaux）功能的变量$\sum_{j=1}^k x_j\,e_j$，其中$x_1,x_2,\ldots,x_k$是真正的，并获得建设性的描述，所有提到的功能，通过全纯函数的复变。从这一描述可以得出，单演函数具有所有阶的加托导数。本文推广了文[1]中关于$k=3$的结果。
英文摘要：
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 obtain a constructive description of all mentioned functions by means of holomorphic functions of complex variables. It follows from this description that monogenic functions have Gateaux derivatives of all orders. The present article is generalized of the author's paper [1], where mentioned results are obtained for $k=3$.
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