Integral theorems in finite-dimensional commutative algebra

Abstract

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.