Pukhtaievych R. A constructive description of monogenic functions in harmonic algebras

In the thesis we study algebraic-analytic properties of monogenic (i. e. continuous and differentiable in the sense of Gateaux) functions in some commutative harmonic Banach algebras associated with the three-dimensional Laplace equation, videlicet, in a three-dimensional algebra A2 with a one-dimensional radical and in an arbitrary finite-dimensional semi-simple algebra An. We have established constructive descriptions of all monogenic functions taking values in both the algebra A2 and the algebra An by means of monogenic functions of the complex variable. We have also established an monomorphism between algebras of monogenic functions in the algebra A2 at transition from a harmonic basis to another one. We have proved analogues of the Taylor theorem and the Laurent theorem for monogenic functions taking values in both the algebra A2 and the algebra An, we have made a classification of singularities of these functions. For monogenic functions taking values in the algebra An, we have proved analogues of the Morera theorem and the Cauchy theorem for a curvilinear integral and the Cauchy integral formula. We have also proved an analogue of the Cauchy theorem for a surface integral with a hyperholomorphic function taking values in the algebra An.