Shpakivskyi V. Monogenic functions in associative algebras

The thesis is devoted to development of the functions theory of the hypercomplex variable in finite-dimensional algebras (commutative and non-commutative) and in infinite-dimensional spaces with commutative multiplication develops in the thesis. In particular, we obtain a constructive description of monogenic functions with values in an arbitrary finite-dimensional commutative associative algebra over the field of complex number by means of holomorphic functions of a complex variable. As a consequence, for an arbitrary linear partial differential equation with constant coefficients, we propose a procedure for constructing an infinite-dimensional family of solutions. For the mentioned monogenic functions we prove analogues of classical integral theorems of complex analysis (the Cauchy theorem for a curvilinear and for a surface integrals, Morera theorem). Monogenic functions with values in the topological vector space, which is an extension of some infinite-dimensional commutative associative Banach algebra associated with the three-dimensional Laplace equation, are considered. New classes of monogenic mappings in the algebra of complex quaternions H(C), so-called right-G-monogenic mappings and left-G-monogenic mappings are introduced. The main algebraic-analytic properties of G-monogenic mappings are investigated. The relation between the well-known classes of quaternion differential functions and the Hausdorff analytic functions is established. We study the left-At-hyperholomorphic (belonging to the kernel of Dirac operator) functions in the generalized of Cayley-Dickson algebras. The main result for such functions is an algorithm of construction of such functions.